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UNAC-FIEE PRACTICA N 1 A) CALCULAR LAS SIGUIENTES INTEGRALES: 1.-

MATEMTICA II

xdxSolucin:

xdx !2.-

x2 c 2

(x

5

3)dx

Solucin:5 5 ( x 3)dx = x dx 3dx =

x6 x6 3 dx c = 3x c 6 6

3.-

( x 2)

4.-

(x

2

( x 2 x 1) 3 ! x 6 3 x 5 6 x 4 7 x 3 6 x 2 3 x 1

3x 2 x 7 x 6 6x5 7x 4 3 ( x 3x 6 x 7 x 6 x 3x 1)dx ! 7 2 5 4 2 x 2 x c6 5 4 3 2

5.-

x(3x

2 3 x(3x 5)dx ! (3x 5 x)dx !

6.-

x(ax

2 3 x(ax b)dx ! (ax bx)dx !

7.-

x

2

(x

2) 3 dx Solucin:3

dx ! ( x 3 6 x 2 12 x 8)dx

= x 3 dx 6 x 2 dx 12 xdx 8 dx = x 1) 3 dx

x4 2x 3 6 x 2 8x 4

Solucin:

2

5) dx

Solucin: 3x 4 5 x 2 c 4 4

2

b )dx

Solucin: ax 4 bx 2 c 4 2

(a bx )dx

1

UNAC-FIEE Solucin:2 2 3 x (a bx)dx ! (ax bx )dx !

MATEMTICA II

ax 3 bx 4 c 3 4

8.-

(x

m

x n ) 2 dx

Solucin:

(x

x n ) 2 dx ! ( x 2 2 x

=(x xn )2 x

x 2 m 1 x 2 n 1 2 x m n 1 c 2m 1 m n 1 2 n 1

9.-

dx

2m m n x 2n x 2x 1 x2

1 1 1 2m 2 n m n 2 x 2 )dx dx ! ( x 2 2 x

=

x

1 2m 2

10.- ( x x 1)( x 1)dx Solucin:5

2x 2 ( x x 1)( x 1)dx ! ( x x 1)dx = ( x 1)dx ! xc 5 11.-

xdx 1 x2 Solucin:

Sea:

Q ! 1 x2

dQ ! 2 xdx p dx !

xdx 1 x21

!

xdQ Q 2x1

!

dQ Q

= 2 Q 2 ! 2(1 x 2 ) 2

n

x 2 n )dx

Solucin:

2m

1 2

2x

m n

1 2

1 mn 2

x

2 n

1 2

1 2n 2

c

3 2

dQ 2x 1 2

! Q

dQ

2

UNAC-FIEE

MATEMTICA II

12.-

( x 1)dx x2 2x 3 Solucin:

Q ! x 2 2x 3 dQ ! ( 2 x 2) dx p ( x 1) dx !1

dQ 2

( x 1)dx x2 2x 31

!

1 dQ 1 Q ! 2 Q 2 dQ 2 1

1 = ( 2 Q 2 ) ! ( x 2 2 x 3) 2 2

13.-

x

2

x 1 dx 2x

Solucin: x 2 2x ! Q (2 x 2)dx ! dQ dQ 2 1 ( x 1) 1 dQ 1 dx ! ! ln Q ! ln x 2 2 x 2 x 2x 2 2 Q 2p ( x 1)dx !

14.-

x3 1 x 4 dx Solucin:1 x4 ! Q 4 x 3 dx ! dQ p x 3 dx ! dQ 4

1 dQ 1 1 x3 4 1 x 4 dx ! 4 Q ! 4 ln Q c ! 4 ln 1 x c 15.-

x(5 3x

2 8

) dx Solucin:

5 3 x 2 ! Q p xdx !

dQ 6

1 1 Q9 5 3x 2 Q9 ! Q 8 dQ ! c! c! 6 6 9 54 54

c9

16.-

x .

2 5

7 4 x 3 dx

Solucin:

3

UNAC-FIEE dQ 12 6/5 1 1 5 6/5 5 7 ! Q 1 / 5 dQ ! Q c ! 4 x 3 c 12 12 6 72

MATEMTICA II

7 4 x 3 ! Q p x 2 dx !

17.-

x

2

1 x dx

Solucin: 1 x ! Q p dx ! dQ! Q 1 Q2 1/ 2

dQ ! 2 2 Q 1 1 / 2 dQ Q Q

! Q 5 / 2 2 Q 3 / 2 Q 1 / 2 dQ ! 2x 1 ! 77/2

4x 1 5

5/2

2Q 7 / 2 2 Q 5 / 2 2Q 3 / 2 2 c 7 5 3 3/ 2 2x 1 c 3

3

18.- ( ax b) 2 dx Solucin:dQ a 1 1 2 5/2 ! Q 3 / 2 dQ ! Q c a a 5 2 ! ax b 5 / 2 c 5a ax b ! Q p dx !

19.- x n 1 a bx n dx Solucin: a bx n ! Q p bnx n 1 dx ! dQ 1 1 2 3/ 2 1/ 2 ! Q dQ ! bn 3 Q c bn 2 a bx n 3 / 2 c ! 3bn 20.-

3 ln x dx x

Solucin:3 Lnx ! Q p dx ! dQ x 3 Lnx 2 c Q2 ! QdQ ! c! 2 2

4

UNAC-FIEE

MATEMTICA II

21.-

ln(ln x ) dx x ln x

Solucin: Q ! ln(ln x) dQ !

1 1 ! (ln x)d dx ln x x.ln x ln(ln x) 2 Q2 Q .dQ ! 2 c ! 2 c

22.-

1 x

x dx x

Solucin: Q ! 1 x x dQ ! ! 3 x.dx 2

2dQ 2 dQ 2 2 ! ! ln Q c ! ln 1 x x c 3Q 3 Q 3 3xdx2

23.-

1 x 2 (1 x 2 ) 3

Solucin: 1 x 2 ! Q p xdx ! dQ 2 1 dQ 1 dQ ! ! 3/ 2 1/ 2 2 2 Q 1 Q 1/ 2 QQ dQ 1 Q 1 / 2 ! Y p 1 / 2 ! 2dY Q 2 dY ! ! 2Y 1 / 2 c 2 Y! 2 1 Q 1/ 2

1/ 2

c ! 1 1 x2

1/ 2 1/ 2

c

24.-

x( x

dx 7 1) Solucin:

x 1 dx x dx ! xx 1 x 1 x7 7 7 7

dx x 7 1 x 7 dx ! x x7 1 x x7 1

5

UNAC-FIEEdx x6 7 dx x x 1 1 ! Ln x Ln x 7 1 c 7

MATEMTICA II

!

25.-

x

3

cos x 4 dx Solucin:

Q ! x 4 dQ ! 4 x 3 dx 1 1 dQ 1 4 cos Q 4 ! 4 cos Q.dQ ! 4 senQ c ! 4 senx c

26.- cos 2 xsenxdx Solucin: cos x ! Q p senxdx ! dQ! Q 2 dQ ! Q3 cos 3 x c ! c 3 3

27.-

sen x dx x

Solucin: dx ! 2dQ x ! 2 senQdQ ! 2 cos Q c ! 2 cos x!Qp 28.-

x cSolucin:

x x

2

dx 4 x 13

2

dx dx 1 x2 ! ! arc tg c 2 2 4 x 13 x 2 3 3 3 dx 2 3

29.-

7x! !

Solucin: dx 1 ! 2 7x 3 7

7 x 3 !2 2

7 dx

7x 1 1 arc tg 3 c 7 3

6

UNAC-FIEE

MATEMTICA II

30.-

4x 4x

dx 4 9 Solucin:

dx 1 2dx 1 2x ! ! arc tg c 2 2 2 9 2 2 x 3 6 3

31.-

x

4

x dx 81 Solucin:

x2 xdx 1 2 xdx 1 ! ! arc tg x 4 81 2 x 2 2 9 2 18 9

c

32.-

x x

2

dx 4x 6 Solucin:

2

dx dx 1 x2 ! ! arc tg c 2 2 4x 6 2 2 x 2 2x5 dx 6 x 12

33.-

x

2

Solucin:1 2 x 6 8 x5 dx dx ! 2 2 x 2 6 x 12 x 6 x 12 1 dx 2 x 6 dx 8 ! 2 x 2 6 x 12 2 x 6 x 12 1 1 x 3 ! Ln x 2 6 x 12 8 arc tg c 2 2 2 1 x 3 ! Ln x 2 6 x 12 4arc tg c 2 2

34.-

e

4x

e 2 x dx 6e 2 x 34

Solucin: 1 2e 2 x dx e 2 x dx ! e 4 x 6e 2 x 34 2 e 2 x 32 5 2 !! e2x 3 e2x 3 1 1 1 c ! arc tg arc tg 5 5 c 2 5 10

7

UNAC-FIEEx2 3 x 2 ( x 2 9) dx

MATEMTICA II

35.-

Solucin:2 1 1 2 x 9 x2 ! x2 x2 9 3 3 3 2 2 2 x 3 1 2x x 9 ! 2 2 dx dx ! 2 2 x x 9 3 x x 9 x3 3 ! x2

2

!

1 2x x 9 x 2 x 2 9 x 2 x 2 9dx ! 3 2

dx 1 2 1 2dx x 1 ! 2 2 ! arc tg c x 3 3 3 x 9 3 x

36.-

x x

2

dx 4

Solucin: dx 1 x2 Ln ! c 2 22 x 2 2 dx 2 3 Solucin:

2

37.-

5x 5x

dx 1 ! 2 3 5dx 6x 7

5dx

5x 2

3

2

!

1 1 5x 3 c Ln 5 2 3 5x 3

38.-

x x

2

Solucin: dx dx x 3 2 1 c ! ! Ln 2 2 6x 7 x 3 2 2 2 x 3 2dx 2x

2

39.-

3x 3x!

2

Solucin: 1 dx dx dx ! ! ! 2 2x 3 2 x / 3 3 x 1 / 32 / 32 x 1

2

1 x 2/3 Ln c 2 x

8

UNAC-FIEE

MATEMTICA II

40.-

x x

2

xdx 5x 3 Solucin:

x 1 2 x 5 5 / 2 dx ! dx ! 2 5x 3 2 x 5x 3 1 2 x 5dx 5 dx 2 ! 2 2 x 5 x 3 2 x 5x 3 5 1 dx ! Ln x 2 5 x 3 2 2 2 2 5 13 x 2 2 5 13 x 1 5 1 2 2 2 c ! Ln x 5 x 3 Ln 2 13 2 5 13 x 2 2 2 2 2

!

5 2 x 5 13 1 Ln x 2 5 x 3 Ln 2 4 13 2 x 5 13 dx

41.-

25 x 5

2

Solucin:x5 dx 1 5 x 1 Ln Ln c ! ! 2 x 5 x 25 5 x 10 dx 42.- 5 2x2 Solucin:2

5 2x43.-

dx

2

!

1 2 dx 5 2 2x 22

2

!

5 2x 1 1 c Ln 5 2x 2 2 5

1 (3x 1) 1 (3x 1)dx

dx

Solucin:! 1 3 x 1 1 3x 1 3dx 1 1 12 3x 12 ! 3 21Ln 1 3x 1 ! 6 Ln 2 3x c 3

2

44.-

25 4 x

dx

2

Solucin:

9

UNAC-FIEEdx 5 2x 1 2dx 1 1 52 2 x 2 ! 2 10 Ln 5 2 x c 2

MATEMTICA II

25 4 x45.-

2

!

4 2x x

3dx

2

Solucin:

4 2x x46.-

3dx

2

! 3

dx

2 2 x 22

2

! 3

1 4 2

Ln

2 2 x2 c 2 2 x2

24 2 x

xdx2

x4

Solucin: xdx 28 x 2 2 1 2 xdx 1 1 ! ! Ln 24 2 x 2 x 4 2 28 2 x 2 22 2 2 28 28 x 2 2 47.-

11 4 x 11 4 x2x 3

2x 32

dx Solucin:

2 xdx 3dx 2 11 4 x 11 4 x 2 3 2dx 1 ! Ln 11 4 x 2 2 2 11 2 x 2 42

dx !

11 2 x 3 1 1 Ln c ! Ln 11 4 x 2 2 2 11 4 11 2 x

48.-

cos cos

2

sen 3 x dx 3 x 2 cos 3 x 3

Solucin: sen 3 x sen3 x 1 cos 3 x 1 2 c ! dx ! dx ! Ln 2 2 3 x 2 cos 3 x 3 22 cos 3 x 1 2 cos 3 x 1 2

2

1 cos 3 x 3 ! Ln c 4 cos 3 x 1 49.-

cos cos

senx cos x dx 2 2x 4

Solucin: sen 2 x 1 1 1 senx cos x cos 2 x ! arc tg dx ! 2 2 2 2 2x 4 2 cos 2 x 2 2 2

10

UNAC-FIEE e arctgx x ln( x 2 1) dx 1 x2 Solucin: e arctgx x ln( x 2 1) e arc tgx xLn 2 1 x dx dx ! 2 dx dx 2 2 2 x 1 x 1 1 x x 1 2 2 1 Ln x ! e arc tgx arctgx c 4

MATEMTICA II

50.-

11

UNAC-FIEE PRACTICA N 2 A) CALCULAR LAS SIGUIENTES INTEGRALES: 1.-

MATEMTICA II

sen( x

2

4 x 5)( x 2)dx

Solucin:

sen ( x

2

4 x 5)( x 2) dx = sen x 2 1 dx2 2

u ! x 2 1 du ! 2x 2 dx dQ 1 1 1 2 senQ 2 ! 2 senQdQ ! 2 cos Q c ! 2 cos x 2 1 c

2.- cos( senx x 2 )(2 x cos x)dx Solucin:Q ! senx x 2 dQ ! cos x 2 x dx2

cos Q.dQ ! senQ c ! sensenx x c3.-

tg ( x 2 4 ) x x2 4

dx

Solucin:Q ! x2 4

dQ !

1 2x dx ! 2 x2 42

xdx x2 4

tgQ .dQ ! ln cos Q c ! ln cos x4.- ctg (ln x )dx x

4 c

Solucin:dx x ctgQ .dQ ! ln senQ c ! ln senln x c Q ! ln x dQ !

5.- sec(3 x 5)dx Solucin:Q ! 3 x 5 dQ ! 3dx dQ 1 1 sec Q 3 ! 3 sec Q.dQ ! 3 ln sec Q tgQ c 1 @ sec3x 5dx ! sec3 x 5 tg 3 x 5

c ln 3

12

UNAC-FIEE

MATEMTICA II

2 x cos x dx 6.- sec 2 (sen x x) 2 x Solucin: 2 x cos x cos x dx = sec 2 sen x x 1 dx ( sen x x ) 2 x 2 x cos x dx Q ! sen x x dQ ! 2 x

sec

2

sec

2

Q .dQ ! tgQ c ! tg sen x x c

7.- sec( senx )tg ( senx ) ctgx . cos x dx Solucin: . . . sec senx sec senx tg senx Q ! sec

cos x dx senxcos x senx

senx

dQ ! sec

senx tg senx 1 . 2

c2

2Q 2 Q .2dQ ! 2 Q .dQ ! 2 c ! sec senx 8.-

1 cos 8 x.dx

Solucin:

1 cos 8 x.dx = 1 2 cos 2 4 x 1.dx ! 2 cos 4 xdx sen 4 x c 2 cos 4 x dx ! 2 4

9.-

x sen( x5

6

2) dx

Solucin:Q ! x 6 2 dQ ! 6 x 5 dx dQ 1 1 1 6 senQ 6 ! 6 senQ .dQ ! 6 cos x c ! 6 cos x 2 c

10.-

sec 2 x x dx Solucin:Q! x

dQ !

1 2 x

dx

13

UNAC-FIEE

MATEMTICA II

sec

2

Q .2dQ ! 2 sec 2 Q .dQ ! 2tgQ c ! 2tg x c

11.- sec hxdx Solucin: sec hx !Q ! ex

1 2 2e x ! x ! 2x cosh x e e x e 1 dQ ! e x dxdQ ex dx ! 2 2 ! 2 arctg Q c ! 2arctg e x c 2x e 1 Q 1

sec hx.dx ! 2

12.- (3senh 7 x 8 cosh 7 x ) dx Solucin:

(3senh7 x 8 cosh 7 x)dx = 3 senh7 x.dx 8 cosh 7 xdx! 3 cosh 7 x 8 senh 7 x c 7 7

13.- cosh 2 xdx Solucin: e x e x cosh xdx ! 2 2 2

cosh

2

xdx !!

1 2x e 4

1 ! e 2 x e 2 x 2 ; reemplazando en la integral dada: 4 1 e 2 x e 2 x e 2 x 2 dx ! 2x c 4 2 2

1 senh 2 x 2 x c ! 1 senh 2 x x c 4 2 4

14.- senh 4 x cosh xdx Solucin:4 senhx cosh x.dx ! senhx cosh x.dx ! 4

senh 5 x c 5

15.- e x cosh( e x ) senh (e x ) dx Solucin:x x x x senh x e x dx . e coshe e dx ! senhe coshe ! Q dQ

senh 2 e x c 2

14

UNAC-FIEE 16.- senh( x ) dx x Solucin:

MATEMTICA II

senh(17.-

x)

dx ! 2 senh x d x ! 2 cosh x

x c

xe

x2

dx Solucin:

Q ! x 2 dQ ! 2 x.dx 1 Q 1 Q 1 x2 Q dQ e 2 ! 2 e dQ ! 2 e c ! 2 e c

18.- sen 2 xe 3cos 2 x dx Solucin:Q ! 3 cos 2 x

dQ ! sen2 x.2dx3 cos 2 x

e19.-

Q

dQ e e ! c ! 2 2 2

Q

c

e x e 2 x e3x e 4 x dx Solucin:e3x e2x ex e x e 2 x e3x dx ! 4 x dx 4 x dx 4 x dx e4x e e e 3 x 2 x ! e dx e dx e x dx

20.-

3e

x

x

dx

Solucin:3ex

x

dx ! 3

e

x

x

dx

Q ! x dQ !

1 dx x 3 e Q .du ! 3e Q k ! 3e x k

21.-

e

x

dx 3

Solucin:

15

UNAC-FIEE 1 3dx 1 x 3 ex 1 ex 3 ex dx e ! x ! dx ! x dx x ex 3 3 e 3 3 3 e 3 e 3 e x 3 1 e x .dx ! dx x 3 e 3 x Q ! e 3 dQ ! e x .dx dQ 1 1 1 ! x ln k ! x ln e x e dx 3 3 3 Q

MATEMTICA II

22.- 5 32 x dx Solucin: dQ 1 5Q 1 5 3 2 x 1 ! 5 Q .dQ ! c ! ln 5 ln 5 c 2 2 2 2 Q ! 3 2 x dQ ! 2.dx3 2 x 5 dx ! 5Q

23.- ( sen 3 x )5 3 4 cos 3 x dx Solucin:

sen3x.5Q 5

3 4 cos 3 x

.dx

Q ! 3 4 cos 3 x dQ ! 4 sen3 x dx ! 12.sen3 x.dx 3 1 5 3 4 cos 3 x dQ 1 1 5Q ! 5 Q .dQ ! c ! ln 5 12 ln 5 c 12 12 12 2 x

24.- e 2 x 7 e dx Solucin:2 x

e

2 x

.7 e .dx2 x

Q ! e 2 x 2 dx ! 2e 2 x dx dQ 1 7Q 1 7e 1 ! 7 Q dQ ! c ! c 7 2 2 2 ln 7 2 ln 7 Q

25.-

a 2 x 1 a x dx

Solucin: 2 x a2x 1 a x dx ! a 2 .dx a 2 .dx ! a 2 dx a 2 .dx 3x 3 x dx Q ! dQ ! dx ; v ! dv ! 2 2 2 2 x x 3x x

16

UNAC-FIEE 2 2 dQ a v 2 dv ! a Q .dQ 2 a v .dv 3 3 x 32x a a 2 av 2 aQ ! 2 c ! 2 c 3 ln a ln a 3. ln a ln a

MATEMTICA II

a

Q

26.- sen (35 x ) dx Solucin:

sen(35 x).dx ! sen35 x 35 dx ! 35 cos 35 x c27.- cos( 43 x) dx Solucin:

35

1

cos(43x)dx ! cos43x 43 dx ! 45 sen43x c28.- sen 2 (5 x)dx Solucin:dx cos 10 x 10 dx 2 2 2 10 1 1 sen10 x 1 ! x sen10 x c ! x c 2 20 2 10

43

1

sen

2

(5 x ) dx !

1 cos 10 x dx !

29.- cos 4 (3 x)dx Solucin: 2 cos 6 x 1 dx 4 4 1 1 cos 12 x 1 ! cos 2 6 xdx 2 cos 6 xdx dx ! dx 2 cos 6 xdx dx 4 4 2 dx dx 1 1 sen12 x x sen 6 x ! cos x 2 cos 6 x dx dx ! 12 c 2 3 4 2 2 4 244 cos (3x)dx !

cos 6 x 12 dx !

cos 6 x

2

30.- sen 29 ( 7 x ) cos (7 x ) dx Solucin:29 sen (7 x) cos(7 x)dx !

sen30 7 x c 210

31.-

cos

17

(9 x ) sen (9 x ) dx17

UNAC-FIEE Solucin:17 cos (9 x)sen(9 x)dx !

MATEMTICA II

cos18 9 x c 162

32.-

sen3

3

(4 x ) dx

Solucin:

sen (4 x)dx ! sen

2

4 x.sen4 x.dx ! 1 cos 2 4 x sen4 x dx .cos 4 x cos 3 4 x c 4 12

! sen 4 xdx cos 2 4 x.sen 4 xdx !

33.- cos 5 3 xdx Solucin:1 cos 3xdx ! cos 3x. cos 3xdx ! sen 3x cos 3xdx ! 2 sen 3x sen 3 x cos 3 xdx 1 ! cos 3 xdx 2 sen 3 x. cos 3 xdx sen 3 x. cos 3 xdx5 4 2 2 2 4 2 4

!

sen3 x 2 sen 3 3 x sen 5 3 x c 3 9 15

34.- tg 2 (3 x ) dx Solucin:

tg (3x)dx ! sec 35.- ctg (5 x ) dx2 4 4

2

dx 3 x 1 !

tg 3 x xc 3 Solucin:

dx ctg (5 x)dx ! ctg 5 x .ctg 5 xdx ! ctg 5 xcos ec 5 x 1 ! ctg 5 x cos ec 5 x dx ctg 5 xdx . ctg 5 x ctg 4 x ctg 4 x cos ! ec 5 x 1 ! dx xc2 2 2 2 2 2 2 3 3 2

15

12

4

36.- tg 6 (5 x ) dx Solucin:

tg

6

(5 x)dx ! tg 4 5 x.tg 2 5 xdx ! tg 4 5 x sec 2 5 x 1 dxtg 5 5 x tg 2 5 x sec 2 5 x 1 dx 25

! tg 4 5 x. sec 2 5 x tg 4 5 xdx !

18

UNAC-FIEE tg 5 5 x tg 2 5 x. sec 2 5 xdx 25 tg 2 5 x tg 3 5 x tg 5 5 x tg 3 5 x tg 5 x ! 2 5 x 1 ! sec dx xc 25 15 25 15 5

MATEMTICA II

!

37.- tg 3 (5 x)dx Solucin:

tg!

3

(5 x ) dx ! tg 2 5 x.tg 5 xdx ! sec 2 5 x 1 tg 5 xdx

tg 2 5 x ln sec 5 x c 10 5

38.- ctg 5 3 xdx Solucin:

ctg

5

3 xdx ! ctg 4 3 x.ctg 3 x.dx ! cos ec 2 3 x 1 ctg 3 x.dx ! csc 4 3 x 2 csc 2 3 x 1 ctg 3 x.dx ! csc 3 3 x. csc 3 x.ctg 3 x.dx 2 ctg 3 x. csc 2 3 x.dx ctg 3 x.dx

2

csc 4 3 x ctg 2 3 x ln sen3 x c 12 3 3 39.- cos 3 xsen 4 xdx!

Solucin:

cos

3

xsen 4 xdx ! cos 2 x.sen 4 xdx ! 1 sen 2 x sen 4 x cos xdx

1 1 ! sen 4 x cos xdx sen 6 x cos xdx ! sen 5 x sen 7 x c 5 7 2 2 40.- sen x cos xdx

Solucin:

sen

2

x cos 2 xdx !

1 1 cos 2 x 1 cos 2 x dx ! 1 cos 2 2 x dx . 2 4 2 sen4 x 1 1 1 cos 4 x 1 dx ! x ! sen 2 2 x.dx ! c 4 4 2 8 4

41.- sen 5 x cos 2 xdx Solucin:

sen

5

x cos 2 xdx ! sen 4 x. cos 2 x.senxdx ! 1 cos 2 x cos 2 x.senxdx ! 1 2 cos 2 x cos 4 x cos 2 x.senxdx ! cos 2 x.senxdx 2 cos 4 x.senxdx cos 6 x.senxdx

2

19

UNAC-FIEE cos 3 x 2 cos 5 x cos 7 x c 3 5 7 Solucin:

MATEMTICA II

!

42.- sen 4 x cos 2 xdx

1 cos 2 x 1 cos 2 x 4 2 sen x cos xdx ! 2 2 dx 1 1 ! 1 cos 2 2 x cos 2 x dx ! sen 2 2 x cos 2 x dx 1 1 8 8

2

!

sen 3 2 x 1 1 cos 4 x 1 sen 2 2 xdx sen 2 2 x. cos 2 xdx ! dx c 8 2 6 8 1 x senx sen 3 2 x ! c 8 2 8 6 Solucin:

?

A

43.- cos 7 x.sen 3 xdx

cos

7

x.sen 3 xdx ! cos 7 x.sen 2 x.senxdx ! cos 7 1 cos 2 x senxdx cos 8 x cos 10 x c 8 10

! cos 7 x.senxdx cos 9 x.senxdx !

44.-

sen

2

3 x cos 4 3 xdx Solucin:

1 cos 6 x 1 cos 6 x sen 3x cos 3xdx ! 2 2 dx 2 4

2

!

1 1 2 2 1 cos 6 x 1 cos 6 x dx ! 8 sen 6 x1 cos 6 x dx 8 1 ! sen 2 6 xdx sen 2 6 x cos 6 xdx 8 sen 3 6 x 1 1 cos 12 x ! dx c 8 2 18

?

A

x sen 2 x sen 3 6 x 1 x sen12 x sen 3 6 x c ! ! c 8 2 24 18 16 192 144

45.- sec 4 ( 2 x ).tg 2 ( 2 x ) dx Solucin:

sec

4

2 x.tg 2 2 xdx ! sec 2 2 x.tg 2 2 x. sec 2 2 xdx ! 1 tg 2 2 x tg 2 2 x. sec 2 2 xdx20

UNAC-FIEE

MATEMTICA II

! tg 2 2 x. sec 2 2 xdx tg 4 2 x. sec 2 2 xdx !

tg 3 2 x tg 5 2 x c 6 10

46.-

tgx . sec 6 xdx Solucin:

tgx sec 6 xdx ! tg 1 / 2 x. sec 4 x. sec 2 xdx ! tg 1 / 2 x 1 tg 2 x sec 2 xdx

2

! tg 1 / 2 x. sec 2 xdx 2 tg 5 / 2 x. sec 2 xdx tg 9 / 2 x. sec 2 xdx ! 2tg 3 / 2 x 4tg 7 / 2 x 2 11 / 2 tg x c 3 7 11

47.- tg 3 3 x sec 3 3 xdx Solucin:

tg

3

3 x. sec 3 3 xdx ! tg 2 3 x. sec 2 3 x.tg 3 x. sec 3 xdx ! sec 2 3 x 1 sec 2 3 x.tg 3 x. sec 3 xdx

! sec 4 3 x.tg 3 x. sec 3 xdx sec 2 3 x.tg 3 x. sec 3 xdx ! sec 5 3 x sec 3 3 x c 15 9

48.- ctg 5 x cos ec 4 xdx Solucin:

ctg

5

x. cos ec 4 xdx ! ctg 5 x. cos ec 2 x. cos ec 2 xdx ! ctg 5 x 1 ctg 2 x cos ec 2 xdx ! ctg 5 x. cos ec 2 xdx ctg 7 x. cos ec 2 xdx !

ctg 6 x ctg 8 x c 6 8

49.-

sen

2

dx x cos 4 x Solucin:

dx sen 2 x cos 2 x 1 1 dx ! ! dx 2 4 4 2 2 sen 2 x. cos 4 x sen x cos x cos x sen x cos x !

sen 2 x cos 2 x dx dx ! sec 4 xdx dx sen 2 x. cos 2 x sen 2 x cos 2 x cos 4 x21

UNAC-FIEE

MATEMTICA II 1 1 ! 1 tg 2 x sec 2 xdx dx 2 2 cos x sen x

! sec 2 xdx tg 2 x. sec 2 xdx sec 2 xdx cos ec 2 xdx ! tgx dx senx cos 3 x

tg 3 x tg 3 x tgx ctgx c ! 2tgx ctgx c 3 3

50.-

Solucin:

!

dx senx cos 3 x

!

sec 2 xdx sec 2 x senx. cos 3 x

!

sec 2 xdx senx. sec 4 x. cos 3 x

sec 2 xdx sec 2 xdx ! ! tg 1 / 2 x. sec 2 xdx ! 2 tgx c senx. sec x tgx

51.-

cos xdx3

sen 7 2 x cos x Solucin:

cos xdx3

sen 7 2 x cos x

!

43 2 3 sen 7 x. cos 8 x 43 2 sec 5 x 3 sen 7 x. cos 8 x tg 2 x sec 2 xdx 1 sec 4 xdx 1 1 ! 3 ! 3 7 tg 3 / 7 x 4 2 3 tg x 4 2! ! ! 1 43

cos xdx

1

sec 5 x. cos xdx

? tg 2

7 / 3

x. sec 2 xdx tg 1 / 3 x. sec 2 xdx

A

1 3 4 / 3 3 x tg 2 / 3 x c 4 tg 2 4 2 1 3 3 ! 3 ctg 4 / 3 x tg 2 / 3 x c 4 2 4 2 3

52.- sen 2 x.sen9 xdx Solucin: 1 sen7 x sen11x c 7 11

sen2 x.sen9 xdx ! 2 cos 7 x cos 11x dx ! 2 53.- cos 2 x cos 7 xdx Solucin:

1

22

UNAC-FIEE 1

MATEMTICA II 1 sen5 x sen9 x c 5 9

cos 2 x. cos 7 xdx ! 2 cos 5x cos 9 x dx ! 2 54.- sen4 x cos 5 xdx Solucin:1

sen 4 x. cos 5 xdx ! 2 sen9 x senx dx ! 2 cos x 55.- sen 3 4 x cos 2 7 xdx Solucin:

1

cos 9 x c 9

sen!

3

4 x cos 2 7 xdx !

1 5sen4 x 3sen10 x sen12 x 2sen18 x sen26 dx c 16

1 3 cos 10 x cos 12 x cos 26 x 5 cos 4 x cos18 x c 16 10 12 26 4 9

56.- sen(3 x 6) cos(5 x 10)dx Solucin:

sen(3x 6). cos(5 x 10)dx ! 2 ?sen8 x 16 sen2 x 4Adx! cos 8 x 16 cos 2 x 4 c 16 4

1

57.- cos x cos 2 5 xdx Solucin:

cos x cos

2

5 xdx !

1 2 cos x cos 9 x cos 11x dx 4 sen9 x sen11x 1 ! 2senx c 9 11 4

T T 58.- sen x sen x dx 4 4

Solucin:

sen 4 x sen 4 x dx ! 2 cos 2 xdx ! 59.-

T

T

1

sen2 x c 4

senx.sen2 x.sen3xdxSolucin:

23

UNAC-FIEE

MATEMTICA II 1

senx.sen2 x.sen3xdx ! 4 sen2 x sen4 x sen6 x dx! 1 cos 6 x cos 4 x cos 2 x cos 2 x cos 4 x cos 6 x c c! 4 6 4 2 8 16 24

60.- senx.sen3 x.sen5 xdx Solucin:

senx.sen3x.sen5 xdx ! 2 cos 2 x cos 4 x sen5x.dx! 1 cos 2 x.sen5 x cos 4 x.sen5 x dx 2 1 1 1 ! sen7 x sen3 x sen9 x cos x dx 2 2 2 1 ! sen 7 x sen3 x sen9 x cos x

dx 4 1 cos 7 x cos 3 x cos 9 x ! senx c 4 7 3 9

1

24

UNAC-FIEE

MATEMTICA II

PRCTICA N 3 1. POR EL MTODO DE INTEGRACIN POR PARTES CALCULAR

A)

xsenxdx ! x cos x cos xdx ! x cos x senx ku ! x du ! dx dv ! senxdx v ! cos x

B)

x

2

senxdx ! x 2 cos x 2 x cos xdx ! x 2 cos x 2 xsenx senxdx ! cos x 2 x 2 2 xsenx ku ! x du ! dx dv ! cos xdx v ! senx

?

A

u ! x 2 du ! 2 xdx dv ! senxdx v ! cos x C) D)

x x

3

senxdx ! senx 3 x 2 6 cos x x 3 6 x k senxdx ! x n cos x n x n 1 cos xdx

n

u ! x n du ! nx n 1 dx dv ! senxdx v ! cos x E)

senx cos xdx ! sen

2

x senx cos xdx !

1 sen 2 x k 2

u ! senx du ! cos xdx dv ! cos xdx v ! senx

F)

x cos xdx ! xsenx senxdx ! xsenx cos x ku ! x du ! dx dv ! cos xdx v ! senx

G)

x

2

cos xdx ! x 2 senx 2 xsenxdx ! x 2 senx 2 x cos x cos xdx ! senx x 2 2 2 x cos x k

?

A

u ! x 2 du ! 2 xdx dv ! cos xdx v ! senx H)

u ! x du ! dx dv ! senxdx v ! cos x

x

3

cos xdx ! cos x 3 x 2 6 senx x 3 6 x k

25

UNAC-FIEE

MATEMTICA II

I)

x

n

cos xdx ! x n senx n x n1 senxdx

u ! x n du ! nx n1 dx dv ! cos xdx v ! senx J)

xsenx cos xdx ! 4 x cos 2 x 4 cos 2 xdx ! 4 x cos 2 x 8 sen 2 x ku ! x du ! dx 1 dv ! senx cos xdx v ! cos 2 x 4

1

1

1

1

2. CON LA INTEGRACIN POR PARTES DEDUCIR LA FRMULA

sen

2

1 1 x 1 xdx ! xsen 2 x xsen2 xdx ! xsen 2 x x cos 2 x cos 2 xdx ! sen2 x k 2 2 2 4u ! x du ! dx dv ! sen 2 xdx v ! 1 cos 2 x 2

u ! sen 2 x du ! sen2 xdx dv ! dx v ! x

3. INTEGRANDO POR PARTES DEDUCIR LA FRMULA

sen

n

xdx ! sen

n 1

x cos x n 1sen

n 2

sen n1 x cos x n 1 n 2 xcos x dx ! sen xdx n n sen x 122

u ! sen n1 x du ! n 1sen n 2 x cos xdx dv ! senxdx v ! cos x 4. CON LOS RESULTADOS DE 2 Y 3 DEDUCIR LAS SIGUIENTES FRMULAS A)

sen

3

1 2 1 3 xdx ! sen 2 x cos x senxdx ! cos 3 x cos x k 3 3 12 4 13

B)

sen sen xdx ! 4 sen x cos x 4 sen xdx ! 8 sen2x x 8 x 16 sen2x ! 8 x 4 sen2x 32 sen4x k4 2 2 1cos 2 x 2

3

1

3

3

3

1

1

1 5 1 4 3 4 1 cos2x 1 1 5 C) sen xdx! sen4 x cosx sen3 xdx! cosx cosx cos3x ! x cos3x cos5x k 80 12 5 4 5 2 5 5 8 485

2

26

UNAC-FIEE

MATEMTICA II

5. CON LA INTEGRACIN POR PARTES Y LOS RESULTADOS DE 2 Y 4 DEDUCIR A)

xsen

2

1 1 1 1 1 1 1 xdx ! x x sen2 x x sen2 x dx ! x 2 xsen2 x cos 2 x k 8 4 4 4 4 2 2

u ! x du ! dx 1 1 dv ! sen 2 xdx v ! x sen2 x 4 2 B)

xsen xdx! 4 x cosx 12 x cos3x 4 cosx 12cos3xdx ! 4 x cosx 12 x cos3x 4 senx 36sen3x k 3

3

1

3

1

3

1

3

1

C)

x

2

x 1 x 1 sen 2 xdx !x 2 sen2 x 2 sen2 x xdx ! 2 4 2 42

x sen xdx ! 6 x2

1

3

1 1 1 1 1 1 1 x 2 sen2 x x cos 2 x cos 2 xdx ! x 3 sen2 x 2 x 2 x cos 2 x k 1 4 2 2 2 8 4 6

6. INTEGRANDO POR PARTES DEDUCIR LA FRMULA RECURRENTEn n 1 n2 sen 2 cos xdx ! senx cos x n 1cos x x dx !

cos x 12

cos n1 xsenx n 1 n 2 cos xdx n n

u ! cos

n 1

x du ! n 1cos

n2

xsenxdx

dv ! cos xdx v ! senx 8. UTILIZANDO 6 DEDUCIR LAS FRMULAS A)

cos

2

xdx !

cos xsenx 1 1 1 cos 0 xdx ! sen 2 x x k 2 2 4 2

1 3 1 cos 2 xsenx 2 B) cos xdx ! cos xdx ! senx sen 3 x ! senx sen3 x k 3 4 12 3 33

C)

4 cos xdx !

1 3 1 cos 3 xsenx 3 cos 2 xdx ! x sen2 x sen4 x k 32 8 4 4 4

9. INTEGRANDO POR PARTES DEMOSTRAR

1 x 2 dx ! x 1 x 2

x2 1 x2

dx

27

UNAC-FIEEx 1 x2

MATEMTICA II

u ! 1 x 2 du ! dv ! dx v ! x

dx

10. DEDUCIR LAS SIGUIENTES FRMULAS A.

a

2

x 2 dx ! x a 2 x 22 n

n

2n x an 2

2

x2

n 1

dx !x a 2 x 2

2n ?a n

2

x2 a2 a2 x2

A

n 1

dx

2 2 2 a x dx ! xa x 2a n a 2 x 2 n1 dx k 2n 1 2n 1 n 2

u ! a2 x2

du ! 2nx an

2

x2

n 1

dx

dv ! dx v ! x

u ! sen n x du ! nsen n 1 x cos xdx 1 senx dv ! dx v ! 1 cos x cos x C. cos 1 x cos x cos 1 x dx ! dx k sen n1 x nsen n x n sen n 1 x u ! cos m x du ! m cos m 1 xsenxdx dv ! cos x 1 dx v ! n 1 sen x nsen n xn

n ln n 1 x u ! ln x du ! dx x x m 1 dv ! x m dx v ! m 1n

E.

2 n n 1 n 2 sen xdx ! sen x cos x n 1sen xcos xdx ! 1 sen x 2

D.

x ln x dx !

1

ln x n

x

1

n n 1 x ln x dx k 1

B.

sen n 1 x sen n x n dx ! cos 1 x cos x

sen n 1 x cos 1 x dx k

sen n1 x cos x n 1 n 2 sen xdx k n n

28

UNAC-FIEE u ! sen n1 x du ! n 1sen n 2 x cos xdx dv ! senxdx v ! cos x F.n n 1 n 2 sen 2 cos xdx ! senx cos x n 1cos x x dx ! 1 cos x 2

MATEMTICA II

cos n 1 xsenx n 1 n 2 cos xdx k n n

u ! cos n 1 x du ! n 1cos n 2 xsenxdx dv ! cos xdx v ! senx G.

sec

n

xdx ! sec

n 2

xtg n 2 sec

n 2

sec n 2 xtgx n 2 n 2 x tg x dx ! sec xdx k ] n 1 n 1 sec 122

u ! sec n 2 x du ! n 2 sec n 2 xtgxdx dv ! sec 2 xdx v ! tgx

H.

x

n

e x dx ! x n e x n x n 1e x dx k

u ! x n du ! nx n 1 dx dv ! e x dx v ! e x

I.

e ax senbx b ax e ax senbx b e ax cos bx b ax e senbxdx ! a a e cos bxdx ! a a a a e senbxdx ! ax

a2 b2 a2

ax e ax senbx be ax cos bx e ax e senbxdx ! ?b cos bx asenbx A k ! 2 a a2 a b2 u ! cos bx du ! bsenbxdx dv ! e ax dx v ! e ax a

u ! senbx du ! b cos bxdx dv ! e ax dx v ! e aax

J.

ax e cos bxdx !

e ax cos bx b ax e ax cos bx b e ax senbx b ax e senbxdx ! e cos bxdx ! a a a a a a

a2 b2 a2

ax e ax cos bx be ax senbx e ax e cos bxdx ! ?bsenbx a cos bx A k ! 2 a a2 a b2

29

UNAC-FIEEu ! cos bx du ! bsenbxdx e dv ! e dx v ! aax ax

MATEMTICA IIu ! senbx du ! b cos bxdx e ax dv ! e dx v ! aax

11. CALCULAR LAS SIGUIENTES INTEGRALES A)

x ln xdx !

1 1 x2 1 1 2 x ln x dx ! x 2 ln x x 2 k 4 2 x 2 2

u ! ln x du !

dx x x2 dv ! xdx v ! 2 1 1 x3 1 1 3 x ln x dx ! x 3 ln x x 3 k 9 3 x 3 3

B)

2 x ln xdx !

u ! ln x du !

dx x x3 dv ! x 2 dx v ! 33

1 1 x4 1 1 4 C) x ln xdx ! x ln x dx ! x 4 ln x x 4 k 16 4 x 4 4 u ! ln x du ! dx x x4 dv ! x 3 dx v ! 4n

x n 1 x n 1 x n 1 x n 1 D) x ln xdx ! dx ! ln x ln x k n 1 xn 1 n 1 n 12 u ! ln x du ! dx x x n 1 dv ! x n dx v ! n 1 x n 1 x n 1 x n 1 x n 1 dx ! ln ax ln ax k n 1 xn 1 n 1 n 1230

E)

n x ln axdx !

UNAC-FIEE

MATEMTICA II

u ! ln ax du !

dx x x n 1 dv ! x n dx v ! n 1

F)

ax xe dx !

xe ax e ax xe ax 1 ax e dx ! 2 k a a a a

u ! x du ! dx e ax dv ! e dx v ! aax

x 2 e ax 2 x 2 e ax 2 xe ax 1 ax x 2 ax 2 x ax 2 ax ax xe dx ! e dx ! e 2 e 3 e k G) x e dx ! a a a a a a a a a2 ax

u ! x 2 du ! 2 xdx dv ! e ax dx v ! e ax a

H)

3 ax x e dx !

x 3 e ax 3 2 ax x 3 e ax 3 x 2 e ax 2 2 x e dx ! 2 xe ax 3 e ax k a a a a a a a

u ! x 3 du ! 3 x 2 dx dv ! e ax dx v ! e ax a

I)

n ax x e dx !

x n e ax n n 1 ax x e dx a a

u ! x n du ! nx n 1dx e ax dv ! e dx v ! aax

J)

ln x dx ! zn

n

e z dz ! e z z n n e z z n 1 dz ! x ln n x n ln x dx

n 1

31

UNAC-FIEEz ! ln x x ! e z dx ! e z dz u ! z n du ! nz n 1 dz dv ! e z dz v ! e z

MATEMTICA II

K)

ln ax dx ! a zn

1

n

e z dz !

1 z n n z n 1 n 1 e z e z dz ! x(ln n ax ) n ln ax dx a a

z ! ln ax ax ! e z dx !

ea dz a

MISCELANEA CALCULAR LAS SIGUIENTES INTEGRALES 1.

lna

2

x

2

dx ! x lna

2

x

2

2

x2 x dx ! x ln a 2 x 2 2 x 2aarctg k 2 2 x a a

u ! ln a 2 x 2 du ! dv ! dx v ! x 2.

2x dx x a22

x 1 ln x 1 dx ! 2 x 1 lnx 1 4 x 1 k dx ! 2 x 1 ln x 1 2 x 1 x 1dx x 1

u ! ln x 1 du ! dv !

dx v ! 2 x 1 x 1 1 x dx ! x ln x 1 x2

3.

x ln

2

x x 1 x2 x2 1 x2 dx 2 2 x 1 x 2 x 2 1 x 1 x x 1 2

2 2 ln x 1 x dx ! x ln x 1 x

x2

1 x2 x 1 x2

x2

dx

32

UNAC-FIEE

MATEMTICA II

u ! ln x 1 x 2 du ! dv ! dx v ! x

x

1 x2 1 x2

1 x x dx 1 x x2 2

4.

x x3 dx !

3x x 3x 3x x 3x dx ! 2 k ln 3 ln 3 ln 3 ln 3

u ! x du ! dx dv ! 3 x dx v ! 3x ln 3

5.

3 3 x2 2 2 dx ! 2 x 2 1 x 4 x 1 x dx ! 2 x 2 1 x 4 x x 2 x 2 dx ! 1 1 3 1 x 3

3 5 x2 16 8 dx ! 2 x 2 1 x x x 2 x 2 k 1 1 15 3 1 x

u ! x 2 du ! 2 xdx dv !x2

u ! x du ! dx dv ! 1 x dx v !3 2 1 x 2 3

dx v ! 2 1 x 1 xx 1 1 x

6.

x dx ! 2 x 2 x dx ! 2 x 2 arctgx k 1 1 1 12 2 2 2 2

1

u ! x du ! dx dv ! x

x 1x3

2 2

dx v !

1 2 1 x2

2 1 x2 3

7.

1 x2

dx ! x 2 1 x 2 2 x 1 x 2 dx ! x 2 1 x 2

3 2

k

u ! x 2 du ! 2 xdx dv ! xdx 1 x2

v ! 1 x2

8.

e

x

dx ! 2 ln zdz ! 2 we w dw ! 2we w 2e w ! 2 x e

x

2e

x

k

33

UNAC-FIEE2 ln z dz z

MATEMTICA II

z!e

x

ln z !

x ln 2 z ! x dx !

w ! ln z e w ! z dz ! e w dw u ! w du ! dw dv ! e w dw v ! e w

9.

xe

2x

dx !

1 2x 1 2x 1 1 xe e dx ! xe 2 x e 2 x k 2 2 2 4

u ! x du ! dx dv ! e 2 x dx v ! e2x 2

10.

x

2

e x dx ! x 2 e x 2 xe x dx ! x 2 e x 2 xe x e x dx ! x 2 e x 2 xe x 2e x k u ! x du ! dx dv ! e x dx v ! e x

?

A

u ! x 2 du ! 2 xdx dv ! e x dx v ! e x

11.

x

2

e 2 x dx !

1 2 2x 1 1 1 1 1 1 x e xe 2 x dx ! x 2 e 2 x xe 2 x e 2 x dx ! x 2 e 2 x xe 2 x e 2 x k 2 2 2 2 4 2 2 u ! x du ! dx dv ! e 2 x dx v ! e 2 x / 2

u ! x 2 du ! 2 xdx dv ! e 2 x dx v ! e 2 x / 2

12.

x e3

x2

dx !

2 2 1 2 x2 1 1 2 x e xe x dx ! x 2 e x e x k 2 2 2

u ! x 2 du ! 2 xdx dv ! xe x dx v ! e x / 22 2

13.

xe tg

1

x 2 3

1 x

dx !

e tg

1

x

1 x

1 2 2

e tg

1

x

1 x

3 2 2

dx

34

UNAC-FIEEe tg x du ! u!e dx 1 x2 1 x dx v ! dv ! 3 1 x2 2 1 x2tg 1 x1

MATEMTICA II

1 2

xe tg

1

x 2 3

1 x u! x

dx !

xe tg

1

x

1 x du !

1 2 2

e tg

1

x

1 x dx

3 2 2

dx

1

x 11

1 2 2

x 11

3 2 2

e tg x dv ! dx v ! e tg 2 1 x

x

IGUALANDO LOS RESULTADOS Y REEMPLAZANDO EN LA PRIMERA ECUACIN

xe tg

1

x 2 3

1 x

dx !

e tg

1

x 1

xtg 1 x 2 1 x2

2 x 2 2 1

14.

x

2

3 x 1 2 x dx ! e

x

2

x 2 3 x 1e 2 x 1 e 2 x 2 x 3 e 2 x dx ! 3 x 1 2 x 1 e 2 x 3e 2 x dx ! 2 2 2 2 2 3x 1 2 x 1 e 1 2 x 3e 2 x e 2 x k 2 4 4u ! 2 x 3 du ! 2 dx dv ! e 2 x dx v ! e 2x 2

x

2

3x 1 2 x dx ! e

x

2

u ! x 2 3x 1 du ! 2 x 3dx dv ! e 2 x dx v ! 15. e2x 2

x x

2

2 x 5 x dx ! 2 2 x 5 x 2 x 1e x dx ! 2 2 x 5 x 2 x 1e x e x dx ! e x e x e 2 x 5 x dx ! 2 2 x 5 x 2x 1e x 2e x k e x e

?

A

2

u ! x 2 2 x 5 du ! 2x 1dx dv ! e x dx v ! e x

u ! x 1 du ! dx dv ! e x dx v ! e x

35

UNAC-FIEE

MATEMTICA II

16.

x x x x x x e senxdx ! senxe e cos xdx ! senxe e cos x e senxdx !

?

A

ex senx cos x k 2

u ! senx du ! cos xdx dv ! e dx v ! ex x

u ! cos x du ! senxdx dv ! e x dx v ! e x

17.

x x x x x x e cos xdx ! cos xe e senxdx ! cos xe e senx e cos xdx !

?

A

ex senx cos x k 2

u ! cos x du ! senxdx dv ! e x dx v ! e x 18. e 2 x cos 3 xdx !

u ! senx du ! cos xdx dv ! e x dx v ! e x

3 3 1 1 3 1 cos 3 xe 2 x e 2 x sen3 xdx ! cos 3 xe 2 x e 2 x sen3x e 2 x cos 3 xdx ! 2 2 2 2 2 2

e

2x

cos 3 xdx !

1 2x e 2 cos 3 x 3sen3 x k 13

u ! cos 3 x du ! 3sen3xdx dv ! e 2 x dx v ! e 2 x / 2

u ! sen3x du ! 3 cos 3 xdx dv ! e 2 x dx v ! e 2 x / 2

19.

e

x

sen 2 xdx ! e x sen 2 x e x sen2 xdx ! u ! sen2 x du ! 2 cos 2 xdx dv ! e dx v ! ex x

u ! sen 2 x du ! sen 2 xdx dv ! e x dx v ! e x

u ! cos 2 x du ! 2sen2 xdx dv ! e x dx v ! e x

x x x x x x e sen2xdx ! e sen2x 2 e cos2xdx ! e sen2x 2 e cos2x 2 e sen2xdx !

?

A

ex sen2x 2 cos2x 5

ex 2 e sen xdx ! 5 5sen x sen2 x 2 cos 2 x kx

2

20. senln x dx ! e z senzdz ! e z senz e z cos zdz ! e z senz e z cos z e z senzdz ! z ! ln x e z ! x dx ! e z dz u ! senz du ! cos zdz dv ! e dz v ! ez z

?

A

x ?senln x cosln x A k 2

u ! cos z du ! senzdz dv ! e z dz v ! e z

36

UNAC-FIEE

MATEMTICA II

21. cosln x dx ! e z cos zdz ! e z cos z e z senzdz ! e z cos z e z senz e z cos zdz ! z ! ln x e z ! x dx ! e z dz u ! cos z du ! senzdz dv ! e dz v ! ez z

?

x A 2 ?senln x cosln xA k

u ! senz du ! cos zdz dv ! e z dz v ! e zx 1

22.

x cos 2 xdx ! 2 sen 2 x 2 sen 2 xdx ! 2 sen 2 x 4 cos 2 x ku ! x dx ! dx dv ! cos 2 xdx v ! x 1 sen 2 x 2 1 x 1

x

1

23.

x cos( 2 x 1)dx ! 2 sen(2 x 1) 2 sen(2 x 1)dx ! 2 sen(2 x 1) 4 cos( 2 x 1) ku ! x du ! dx dv ! cos( 2 x 1) dx v !

1 sen ( 2 x 1) 2 x 1 x 1 24. xsen3 xdx ! cos 3 x cos 3 xdx ! cos 3 x sen 3 x k 3 3 3 9 u ! x dx ! dx 1 dv ! sen3 xdx v ! cos 3 x 3

25.

2 x sen3xdx !

x2 x2 2 xsen3x 1 2 sen3xdx ! cos 3 x x cos 3 xdx ! cos 3x 3 3 3 3 3 3 x2 2 2 cos 3 x k cos 3 x xsen3x 27 9 3u ! x dx ! dx dv ! cos 3 xdx v ! 1 sen3 x 3

2 x sen3xdx !

u ! x 2 dx ! 2 xdx 1 dv ! sen3 xdx v ! cos 3 x 3

26.

x x

2

senhxdx ! x 2 cosh x 2 x cosh xdx ! x 2 cosh x 2 xsenhx senhxdx !

?

A

2

senhxdx ! x 2 cosh x 2 xsenhx 2 cosh x k

37

UNAC-FIEE u ! x 2 du ! 2 xdx dv ! senhxdx v ! cosh x 27.

MATEMTICA IIu ! x du ! dx dv ! cosh xdx v ! senhx

x x

2

cosh xdx ! x 2 senhx 2 xsenhxdx ! x 2 senhx 2 x cosh x cosh xdx ! cosh xdx ! x 2 senhx 2 x cosh x 2senhx ku ! x du ! dx dv ! senhxdx v ! cosh x

?

A

2

u ! x 2 du ! 2 xdx dv ! cosh xdx v ! senhx 28.

xsen xsen

3

1 1 x 3 1 1 cos 2 x xdx ! x dx senxdx ! xsenxdx ?sen x sen3 x A ! xsenxdx xsen3 xdx 2 2 2 2 4 4 3 3 1 1 xdx ! x cos x senx x cos 3 x sen3 x k 4 4 12 362 2 2 2

3

29.

1 1 dx xsenx cosx ! 2 xsen2 x dx ! 8 cos2 x k u ! 1 du ! 0 1 dv ! xsen 2 x 2 dx v ! cos 2 x 2 4

1

30.

sen 2 x cos xdx ! senxsen 2 x 2 senx cos 2 xdx ! 3 ?senxsen 2 x 2 cos x cos 2 x A ku ! sen 2 x du ! 2 cos 2 xdx dv ! cos xdx v ! senx u ! cos 2 x du ! 2 sen 2 xdx dv ! senxdx v ! cos x 1 3

31.

2 3x sen5 xdx ! 2 3x 5 cos 5 x 5 cos 5 xdx ! u ! 2 3 x du ! 3dx 1 dv ! sen5 xdx v ! cos 5 x 3

3x 2 5

cos 5 x

3 sen 5 x k 25

32.

x

2

2 x 3 cos 2xdx !

x

2

2x 3 x 1cos 2 x k 2 x 2 4 x 25 sen2x sen2xx 1dx ! sen2 x 2 4 2

38

UNAC-FIEEu ! x 2 2 x 3 du ! 2 x 2 dx dv ! cos 2 xdx v ! 1 sen 2 x 2

MATEMTICA IIu ! x 1 du ! dx 1 dv ! sen 2 xdx v ! cos 2 x 2

33.

sec

3

1 xdx ! sec xtgx sec x tg 2 x dx ! ? xtgx ln sec x tgx A k sec ] 2 2 sec x 1

u ! sec x du ! sec xtgxdx dv ! sec 2 xdx v ! tgx

34.

5 3 3 2 sec xdx ! sec xtgx 3 sec x tg x dx ! ] sec x 12

sec 3 xtgx 3 ? xtgx ln sec x tgx A k sec 4 8

u ! sec 3 x du ! 3 sec 3 xtgxdx dv ! sec 2 xdx v ! tgx

35.

x sec

2

xdx !xtgx tgxdx ! xtgx ln sec x k

u ! x du ! dx dv ! sec 2 xdx v ! tgx

36.

x sec

2

(ax ) dx !

xtg ax 1 xtg ax 1 tg ax dx ! 2 ln secax k a a a a

u ! x du ! dx dv ! sec 2 ( ax ) dx v ! tg ( ax ) / a sec3 x 3x 1 3 tg 1 tg ? 3x 3x ln sec x tg3xA k sec tg 3 37. sec5 3xdx ! sec3 3x 3x 3 sec3 3xtg 2 3xdx ! 12 8 3 sec 3x12

u ! sec 3 3 x du ! 9 sec 3 3 x tg 3 x dx dv ! sec 2 3 x dx v ! 1 tg 3 x 3

38.

csc

3

1 xdx ! csc xctgx csc x ctg 2 x dx ! ? csc xctgx ln csc x ctgx A k 2 csc x 12

39

UNAC-FIEEu ! csc x du ! csc xctgxdx dv ! csc 2 xdx v ! ctgx

MATEMTICA II

39.

sen

1

xdx ! xsen 1 x dx

x 1 x2

dx ! xsen 1 x 1 x 2 k

u ! sen 1 x du ! dv ! dx v ! x

1 x2

40.

sen ax dx ! xsen ax a 1 1

x 1 ax 2

dx ! xsen 1 ax

1 2 1 ax k a

u ! sen 1 ax du ! dv ! dx v ! x

adx 1 ax 2

_ x 1 x 1 x 1 2 41. sen 1 ax dx ! xsen 1 dx ! xsen 1 1 2 2 2 2 2 x x 2 arcsen x k x 2x x dx u ! sen 1 2 du ! 2 x 2 x dv ! dx v ! x

1 ?2 1 x 2 A 2

42.

xsen sen x dx ! x sen x 2 1

2

1

2

1

x

1 x21

dx ! x sen 1 x

2 sen2

1

x 1 x2

dx ! 1 x2 1 x2

sen x dx ! xsen x 2 sen1

2

1

2

x 1 x 2 2x k

40

UNAC-FIEE

MATEMTICA II 2sen 1 xdx 1 x2

u ! sen 1 x du ! dv ! dx v ! x 43. xsen1xdx !

2

2x 2 1 x 2 sen1x 1 x 1 x2 x 2 sen1 x 1 x2 arcsenx dx ! k x 1 x2 1 x2 dx ! 4 2 2 1 x 2 2 2 4 dx 1 x2

?

A

u ! sen 1 x du ! x2 dv ! xdx v ! 2

u ! x du ! dx dv ! x 1 x2

dx v ! 1 x 2

44.

1 xtg xdx !

x2 1 1 x 2 tg 1 x 1 dx x x 2 tg 1 x 1 x2 dx ! dx ! 2 2 2 arctgx 2 k 2 2 1 x 2 2 2 1 x dx 1 x2

u ! tg 1 x du ! dv ! xdx v !

x2 2

2 2 x 3tg 1 x ln1 x 2 1 x 2 x 3tg 1 x 1 x 3 x 3tg 1 x 1 x ln1 x 2 x ln1 x dx ! k dx ! 45. x tg xdx ! 3 3 1 x 2 3 3 2 3 6 6 21

u ! tg 1 x du !

dx 1 x2 x3 dv ! x 2 dx v ! 3

u ! x 2 du ! 2 xdx ln 1 x 2 x dx v ! dv ! 2 1 x2

ln udu ! uln u 1

46.

1 2 2 x sen 1 x dx !

x 3 sen 1 1 x 2 1 x 3 sen 1 1 x 2 1 x3 dx ! x 2 1 x 2 2 x 1 x 2 dx ! 2 3 3 1 x 3 3

?

A

41

UNAC-FIEE x 3 sen 1 1 x 2 1 x 2 1 x 2 2 1 x2 3 3 93 2

MATEMTICA II

2 2 1 x sen 1 x dx !

k

u ! sen 1 1 x 2 du ! x dv ! x dx v ! 32 3

dx 1 x2

u ! x 2 du ! 2 xdx dv ! x 1 x2

dx v ! 1 x 2

47.

tg

1

xdx ! xtg 1 x

x 1 dx ! xtg 1 x ln 1 x 2 k 2 2 1 x

u ! tg 1 x du ! dv ! dx v ! x 48.

dx 1 x2

tg

1

x dx ! 2 utg 1udu ! x 1arctg x x k

u 2 ! x 2udu ! dx Del _ problema _ 44

49.

ctg

1

xdx ! xctg 1 x

x 1 dx ! xctg 1 x ln 1 x 2 k 2 2 1 x

u ! ctg 1 x du ! dv ! dx v ! x 50.1 xctg xdx !

dx 1 x2

x2 1 x 2 ctg 1 x 1 x2 x 2 ctg 1 x 1 dx x 1 dx ! dx ! 2 2 2 arctgx 2 k 2 2 1 x 2 2 2 1 x dx 1 x2

u ! ctg 1 x du ! dv ! xdx v ! x2 2

51.

sen1 x

x 1

2 3

dx !

z cos z 1 sen2 z 2 cos z 3 2

dz ! z sec2 zdz ! ztgz tgzdz ! ztgz ln sec z !

xsen1 x

1 ln 1 x 2 k 1 x2 2

42

UNAC-FIEEx ! senz dx ! cos zdz u ! z du ! dz dv ! sec 2 zdz v ! tgz

MATEMTICA II

52.

sen1 x 1 x

dx ! 2

zarcsenz 1 z 2 dz ! 2 1 z 2 arcsenz dz ! 2 1 z 2 sen1 z 2z ! 2 1 xsen1 x 2 x k 2 2 1 z 1 z

x ! z 2 dx ! 2 zdz u ! arcsenz dv ! z 1 z2

dz 1 z 2

dz v ! 1 z 2

53.

xsen 1 x 1 x2

dx ! sen x 1 x 1

2

1 x2 1 x2

dx ! sen 1 x 1 x 2 x k

u ! sen 1 x du ! dv ! x 1 x2

dx 1 x2

dx v ! 1 x 2

54.

xtg 1 x 1 x2

dx ! tg 1 x 1 x 2 dx 1 x2

1 x2 dx ! tg 1 x 1 x 2 ln x 1 x 2 k 2 1 x

u ! tg 1 x du ! dv ! x 1 x2

dx v ! 1 x 2 ztg 2 z sec 2 z tg 1 x ln secarctgx k dz ! z tgz z tgz z dz ! xtg 1 x 2 1 tg 2 z2

55.

x 2 tg 1 x dx ! 1 x2

x ! tgz dx ! sec 2 zdz u ! z du ! dz dv ! tg 2 zdz v ! tgz z

43

UNAC-FIEE ctg 1 x 1 1 x dx ! 2 ctg udu ! 2 xctg x ln 1 x kx ! u 2 dx ! 2udu Del _ problema _ 49

MATEMTICA II

56.

57.

senx x

cos x xsenx 12

dx ! dx !

cos xcos x 1

senx x

2

dx

senx senx senx cos x dx ! dx dx ! senx x senx x senx x senx x

senx x

cos x xsenx 12

cos x ! k senx x

u ! cos x du ! senxdx cos x 1 1 dx v ! dv ! 2 senx x senx x 58.

x cos x senx x cos x senxu!x22

x2

2

dx !

xxsenx senx x cos x senx 2

dx !

x senx x cos x dx ! 2 senx x cos x senx sen x x cos x senx

dx !

x ctgx k senxx cos x senx

senx x cos x x dx du ! senx sen 2 x xsenx 1 dx v ! dv ! 2 x cos x senx x cos x senx

x 2sen x cos x x senx 2 2 dx ! 1 x sec 2 x dx tg x dx ! dx ! 59. 2 2 2 1 cos x 2 cos 2 x 2

x x dx x dx x 2 1 cos x dx ! 2 ? xtg 2 2 tg 2 A tg 2 ! xtg 2 k x senx 1

u ! x du ! dx dv ! sec 2 x dx v ! 2tg x 2 2

44

UNAC-FIEE

MATEMTICA II

PRACTICA N4

INTEGRACION Y SUS APLICACIONES PRELIMINARES I) INTEGRALES Y ECUACIONES DIFERENCIALES: A) RESOLVER LAS SIGUIENTES ECUACIONES DIFERENCIALES: 1) y ' ! 2 xy 2dy ! 2xy 2 dx

y

dy2

! 2 x.dx

1 !x y1 x

y!

2) y ! 3x 2

dy ! 3x .dx2

y ! x3

3) y '! x 2 y

y"0

dy ! x 2dx y x3 3

2 y!

y!

x3 645

UNAC-FIEE

MATEMTICA II

4) y ' ! xy

x "0 y "0

dy ! x.dx y 2 3 x 3

2 y!

y!

x3 9 y x x "0 y "0

5) y ' ! 3

3

dy dx !3 2 y x

y ! x

6)

ds ! 3t 2 4t 6 dt2

ds ! 3t

4t 6 dt

s ! t 3 2t 2 6t

7)

dx !8 x dt 1 x dx

x"0

dt ! 8t! 1 x 4

8)

dr 3 ! 2 z 1 dt

46

UNAC-FIEE

MATEMTICA II

dr ! 2 z 1 dz3

r!

2 z 144

9)

2 dy ! t t 1 2 dt1 2

t "0

dy ! 2t t dty ! 4t 2 4 t 2 dt y! 4t 3 1 4t 3 t

10)

dy ! dz

z z2 2

2

z2 42 2

z"0

dy ! y!

z 2 4 .dz2

z

z 2 .dz

y ! z 2 z 2 .dz

2

z3 y ! z 1 3 II) INTEGRALES INDEFINIDAS Y LAS ECUACIONES DIFERENCIALES CON VALORES INICIALES: 1) LA VELOCIDAD DE UN MVIL EN EL INSTANTE T ESTA DADO POR V=AT DONDE A = CONSTANTE. SI LA POSICIN DEL CUERPO EN EL INSTANTE T=0 ES SO, HALLAR S EN FUNCIN DE T:

47

UNAC-FIEEv ! at ds ! v.dt

MATEMTICA II

ds ! at dtat 2 s! k 2 PARA t ! 0 :s !0k k ! s ! so

s!

at 2 so 2

2) HALLAR LA CURVA QUE PASA POR EL PUNTO (1,-1) Y CUYA PENDIENTE EN EL PUNTO (X, Y) EX 3X2 : DE LA CONDICIN SE TIENE:mlt ! dy ! 3x 2 dx

DE DONDE:dy ! 3 x 2 dx

dy ! 3x dx2

y ! x3 k 1 ! 1 k 2!k

SE REEMPLAZA:y ! x3 2

48

UNAC-FIEE

MATEMTICA II

3) HALLASE LA ECUACIN DE LA CURVA CUYA PENDIENTE EN EL PUNTO (X, Y) ES 3X2 + 2, SABIENDO QUE PASA POR EL PUNTO (1,-1): DE LA CONDICIN SE TIENE:ml t ! dy ! 3x 2 2 dx

DE DONDE:dy ! 3 x 2 2 dx

dy ! 3x

2

2 dx

y ! x3 2x k 1 ! 1 2 k 4! k

SE REEMPLAZA:y ! x3 2x 4

4) POR EFECTO DE PRDIDAS, UN CONDENSADOR ELCTRICO SE DESCARGA CON UNA VELOCIDAD PROPORCIONAL A SU CARGA .SI R TIENE EL VALOR RO EN EL INSTANTE T=0 .HALLAR Q EN FUNCIN DE T: 5) EN CADA UNO DE LOS SIGUIENTES PROBLEMAS HALLESE LA FUNCIN S DE LA VARIABLE INDEPENDIENTE T, CONOCIDA LA VELOCIDAD V=DS/DT, ASI COMO LA CONSTANTE DE INTEGRACIN PARA QUE SE TENGA S=SO EN T=0:

A) v !

ds dt

ds ! 3t 2 dt

ds ! 3t dt2

49

UNAC-FIEE s ! t3 kso ! 0 k so ! k s ! t 3 so

MATEMTICA II

B) 2t 1 ! v

ds ! 2t 1dts ! t2 t kso ! k s ! t 2 t so

C) v ! t 1

2

ds ! t 1 dt2

s!

t 13 k3

s!

t 13 s3

o

D) v ! t 2 1t ds ! 2

2

1 dt

2

s!

t 5 2t 3 t k 5 3

t 5 2t 3 s! t so 5 3 E) v ! t 12

50

UNAC-FIEE

MATEMTICA II

ds ! t 1

2

ds

s ! t 1 ks ! t 1 so

F) v ! 2 gsds ! 2 gs .dt

g ! cons tan te

ds ! 2 g .dt s

2 s ! 2 g .t k 2 s !k

k ! 2 sos! 2 g .t 2 so 2

6) EN CADA UNO DE LOS PROBLEMAS SIGUIENTES HALLASE LA VELOCIDAD S QUE DETERMINA LA POSICIN DEL MVIL COMO FUNCIONES DEL TIEMPO T, CUANDO SE CONOCE LA ACELERACIN A=DV/DT.HALLAR LA CONSTANTE DE INTEGRACIN PARA QUE SE TENGA V=VO ,Y S=SO PARA T=0: A) a ! gdv !g dt

g ! cons tan te

dv ! gdtv ! gt kv ! gt vo

51

UNAC-FIEEds ! gt vo dt

MATEMTICA II

ds ! gt v dto

gt 2 s! vot k 2 gt 2 s! vot so 2 B) a ! tdv !t dt dv ! tdt

dv ! tdtt2 v! k 2 v! t2 vo 2

t2 ds ! vo dt 2

s!

t3 vot k 3

t3 s ! vo t so 3 C) a ! 3 2t 1dv 3 ! 2t 1 dt

d dv ! 2t 1 t3

52

UNAC-FIEE33 2t 14 k 8

MATEMTICA II

v!

vo ! 1 k k ! vo 1 v! 33 2t 14 vo 1 8 33 4

ds ! 8 2t 1

vo 1dt

7 3 3 2t 13 s! vo t t k 8 2 7 9 3 s! 2t 17 vot t k 112

so !

9 k 112

s!

9 3 2t 17 vot t so 9 112 1123

D) a ! 2t 1dv 3 ! 2t 1 dt

dv ! 2t 14

3

dt

2t 12 k v!vo ! 1 k 4 2t 12 1 vo dt ds ! 4 4

53

UNAC-FIEE

MATEMTICA II

2t 11 v t t s!4o

4

k

so !

1 k 4

2t 11 v t t s!4o

4

so

1 4

E) a ! t 2 12 dv ! 2 1 t dt

2

t dv !

2

1 dt

2

v!

t 5 2t 3 t k 5 3

vo ! k

t 5 2t 3 v! t vo 5 3 t 5 2t 3 t vo dt ds ! 5 3 t 5 2t 3 ds ! t vo dt 5 3

s!

t6 t4 t2 vot k 30 6 2

so ! k

s!

t6 t4 t2 vot so 30 6 2

F) a ! 2t 1

54

UNAC-FIEEdv ! 2t 1 dt

MATEMTICA II

dv ! v!

2t 1

1 2t 1 k 3 1 k 3

vo !

v!

2t 133

vo

1 3 1 3

ds !

2t 133

vo

ds ! s!

2t 133

1 vo dt 3 t k 3

2t 1515 1 k 15

vo t

so !

s!

2t 1515

vo t

1 t so 3 15

7) RESUELVA LAS SIGUIENTES ECUACIONES CON LAS CONDICIONES INICIALES QUE SE INDICA: A) y ' ! x y PARA x ! 0 y ! 1

dy ! x.dx y

55

UNAC-FIEE x2 2

MATEMTICA II

2 y!

y!

x4 k 16 x4 1 16 PARA x ! 1 y ! 1

y!

B) y ' ! 2 xy 2

y

dy2

! 2 xdx

y 1 ! x 2 k k ! 2

1 ! x2 2 y 1 2 !y x 2

C) y ' ! x 1 x 2

PARA x ! 0 y ! 3

dy ! y!

2x 1 x2 dx 22 3

x k 13 1 k 3

3!

k!

10 3

y!

x 10 12 3

3

3

56

UNAC-FIEE 4 1 y2 y

MATEMTICA II

D) y ' !

3

PARA x ! 0 y ! 1

4 1 y2 dy ! y2 ydy

3

y 11

2 3

! 8dx

y 12 3

! 8x k

k!

1 2

1 1 8x 2

!

1 y 2

1 ! 1 y2 1 8x 2

2

1 2 1 y ! 8x 1 2

2

2 y! 8 2x 1 1

E) y ' ! x x 2 4

PARA x ! 2 y ! 3

dy !

2x x2 4 dx 2

57

UNAC-FIEE

MATEMTICA II3

y!

x x

2

4 k 3 4 4 33

2

y!

F) y ' ! xy 3

PARA x ! 0 y ! 1

y

dy3

! xdx

1 x2 ! k y2 21 ! 2k 9 19 !k 9

1 ! y2 x 19 2 92

y!

1 x 19 2 92

y!

18 9 x 2 38

III) INTEGRACION Y FUNCIONES HIPERBOLICAS (CABLES SUSPENDIDOS): x 1) PROBAR QUE y ! a cosh ES SOLUCIN DE LA ECUACIN DIFERENCIAL: a d2y W dy ! 1 2 dx H dx 2

58

UNAC-FIEE

MATEMTICA II

CON

a!

H W

/H, W SON CONSTANTES.2

d2y W dy ! 1 2 dx H dx dy x ! a cosh dx a d2y x ! asenh 2 dx a

x x W asenh ! 1 sen 2 h a a H

x x asenh ! a 1 sen 2 h a a

x x cos 2 h = 1 sen2 h a a x x cos 2 h sen 2 h ! 1 a a 2) RESOLVER LA ECUACIN DIFERENCIAL: d2y W dy ! 1 2 dx H dx 2

CUYAS CONDICIONES SON:dy ! 0 , y ! yo PARA x ! 0 dx

3) PRUBESE QUE LA TENSIN DEL CABLE EN EL PUNTO P(X, Y) DE LA FIGURA ADJUNTA ESTA DADA POR T=W.Y:

59

UNAC-FIEE

MATEMTICA II

4) LA LONGITUD DEL ARCO AP DE LA FIGURA ANTERIOR ES x y ! asenh .PROBAR QUE LAS COORDENADAS P(X, Y) SE PUEDEN EXPRESAR a COMO FUNCIONES DE LA LONGITUD DEL ARCO S EN LA FORMA SIGUIENTE S y ! aarcsenh , y ! S 2 a 2 a 5) CALCULAR2 2

dx dy Y DEL PROBLEMA ANTERIOR Y COMPRUBESE ds ds

QUE

dx dy !1 ds ds

6) UN CABLE DE 32M DE LONGITUD, CUYO PESO ES 2KG/M, TIENE SUS EXTREMOS FIJOS EN LOS PUNTOS AL MISMO NIVEL SOBRE DOS POSTES SEPARADOS 30M. IV) PROBLEMAS PROPUESTOS: 1) RESOLVER LAS SIGUIENTES ECUACIONES DIFERENCIALES:

A)

dy ! xy 2 dx

dy ! xdx y2 1 x2 ! k y 2

B)

dy ! 1 x y xy dx

dy x 2 1 ! C) dx y 2 1

y

2

1 dy ! x 2 1 dx

y 2 dy dy ! x 2 dx dx

60

UNAC-FIEE y3 x3 y ! xk 3 3 D) dy y y ! dx x x

MATEMTICA II

1 1 2 x x dx ! y y 2 dy

1

1

xdx x 2 dx ! ydy y 2 dyx 2 2x 2 y2 2y 2 ! k 2 3 2 33 3

dy 2 x E) ! dy 3 y

2

dx dy ! 2 x 2 3 y 2

1 1 ! k x 2 y 3

2) UNA PARTCULA SE MUEVE A LO LARGO DEL EJE X CON ACELERACIN A=T2, HALLNDOSE EN EL ORIGEN EN EL INSTANTE T=0.EN EL TRANSCURSO DE SU MOVIMIENTO LA PARTCULA LLEGA AL PUNTO X=B /B>0, PERO NO TRANSPONE B.HALLESE SU VELOCIDAD EN T=0:dv ! t 2 dt

dv ! t dt2

v!

t3 k 3

61

UNAC-FIEE

MATEMTICA II

t4 dx ! k dt 12

x!

t4 k 12

0 4 k b!12b!k

VELOCIDAD EN T=0: t3 v ! b 3 v!

03 b3 PARTCULA SE MUEVE CON UNA ACELERACINa! t 1 , t

3)

UNA

SUPONIENDO QUE V=2 Y S=5 PERA T=0, HALLASE: A) LA VELOCIDAD V EN FUNCIN DE T.

a! t

1 t

dv 1 ! t dt t

dv ! 3

t

1 dt t

2t 2 v! 2t 2 k 31 20 2 20 2 k 2! 3 3

1

2!k62

UNAC-FIEE

MATEMTICA II

3

2t 2 v! 2t 2 2 3 B) EL ESPACIO S EN FUNCIN DE T. ds 2t ! 2t 2 2 dt 3 3 1 2t 2 ds ! 2t 2 2 3 5 3 3 2 1

1

2t 3 4t 2 s! 2t k 5 320 3 40 2 5! 20 k 5 3 5!k5 3

2t 4t 2 s! 2t 5 5 3 4) LA VELOCIDAD DE UNA PARTCULA, SOMETIDA A LA ACELERACIN 3+2T, EN EL INSTANTE T=0 VALE 4.HALLASE SU VELOCIDAD EN FUNCIN DEL TIEMPO Y LA DISTANCIA ENTRE LAS POSICIONES DE LA PARTCULA EN LOS INSTANTES T=0 T=4:dv ! 3 2t dt

5 3

3

dv ! 3 2t dt

dv ! 3 2t dtv ! 3t t 2 k PARA T=0 LA VELOCIDAD ES 4:

63

UNAC-FIEE4 ! 30 0 k 4!k2

MATEMTICA II

SE REEMPLAZA Y SE OBTIENE LA VELOCIDAD EN FUNCIN DEL TIEMPO: v ! 3t t 2 4 HALLAMOS LA DISTANCIA ENTRE POSICIONES DE LA PARTCULA EN LOS INSTANTES T=0 T=4: xt2 xt1 ! x4 x 0 x' t ! vt xt ! vt k dt xt ! 3t t 2 4 dt k xt ! 4t 3t 2 t 3 k 2 3

PARA T=0: 30 0 x0 ! 40 k 2 32 3

PARA T=4: 34 4 x4 ! 44 k 2 32 3

x ! 61.3

DISTANCIA:2 3 2 3 30 0 34 4 44 40 x4 x0 ! k k 2 3 2 3

x ! 61.3

64

UNAC-FIEE

MATEMTICA II

5) LA ATRACCIN EJERCIDA POR LA TIERRA SOBRE UNA PARTCULA DE MASA M A LA DISTANCIA S DEL CENTRO, EST DAD POR F=M.G. R2. S-2, EN DONDE R ES EL RADIO DE LA TIERRA Y F ES NEGATIVA PORQUE ACTA DE FORMA S DECRECE.SI UNA PARTCULA SE LANZA VERTICALMENTE HACIA ARRIBA DESDE LA SUPERFICIE DE LA TIERRA CON VELOCIDAD INICIAL vo ! 2 gR , APLQUESE dv LA SEGUNDA LEY DE NEWTON F=M.A, TAL QUE a ! v , PARA PROBAR QUE ds 3 3 R 3v t Y QUE S 2 ! R 2 1 o . v ! vo 2R s NOTA: vo ! 2 gR =VELOCIDAD DEL ESCAPE EN LA SOLUCIN SE DESPRECIA LA RESISTENCIA OPUESTA POR EL AIRE.F ! mgR 2 s 2 F ! ma ma ! mgR 2 s 2 dv v v ! o ds 2 R2

vdv ! gR sv ! 2 gR 2 s 1 vo ! 2 gR vo ! 2 gR2

2 2

ds DE DONDE:

vo !g 2Rv ! 2 gR 2 s 1

2

REEMPLAZANDO EN:

v!2

vo 2 1 R s 2R R s

2

v ! vo

65

UNAC-FIEE PRCTICA N5

MATEMTICA II

POR SUSTITUCIN TRIGONOMTRICA, CALCULAR LAS SIGUIENTES INTEGRALES: 1.- x 2 5dx

SOL:

tgU !x2 5

x 5

5tgU ! x 5 sec 2 UdU ! dx 5 sec U ! x2 5

I ! 5 secU 5 sec2 UdU ! 5 5 sec3UdU 2.- 7 x 2 dx SOL: 7 senU ! x 7 cosUdU ! dx 7 cosU ! 7 x 2 U sen2U k 7 cosU 7 cosUdU ! 7 cos 2UdU ! 7 4 2 x 7 x 2 arcsen x 7 7 5 k I ! 7 2 2

I!

3.- SOL:

dx x 2 25

66

UNAC-FIEE5 cscU ! x 5 csc UctgUdU ! dx 5ctg ! I ! x 2 25

MATEMTICA II

5 csc UctgUdU ! cscUdU ! ln sec U tgU k 5ctgU

I ! ln sec arc csc x

5 tg cscx 5 k arc

4.-

dx

x3 x 2 9 SOL:3 csc U ! x 3 csc UctgUdU ! dx 3ctgU ! x2 9

I!

27 csc U 3ctgU 3

3 cscUctgUdU

!

U 1 1 1 1 U sen2U 2 2 sen UdU ! 27 dU 27 cos UdU ! 27 27 2 4 k 27

I !

arc csc x

2 3 x 9 arc csc x x 3 1 3 x k 27 27 2 2

5.- SOL:

9 x2 dx x23senU ! x 3 cos UdU ! dx 3 cot U ! 9 x 2

3 cos U .3 cos UdU ! tg 2UdU ! tgU U k 2 9 sen U x x arcsen I ! 2 3 9x I !

67

UNAC-FIEEx2 9 x2 3tgU ! x * 3 sec 2 UdU ! dx * 3 cos U ! 9 x 2 32 tg 2U 3 sec 2 UdU ! 9 tg 2U sec 3 UdU 3 cos U 9tg 2U sec3 U 9 secUtgU ln sec U tgU I ! k 4 4 2 2 I !

MATEMTICA II

6.- SOL:

3 ln 3 3 3 x 2 9x2 x 3 9x 9 3 9 x2 9 x2 I ! k 2 4 4 2

7.-

dx 1 4x2sen U ! 2 x cos UdU ! 2 dx cos U ! 1 4 x 2

SOL:

I!

2 cosU

cosUdU

!

1 dU 2

1 1 I ! U k ! arcsen(2 x ) k 2 2 8.- SOL:dx 4 ( x 1) 2

68

UNAC-FIEE2 senU ! x 1 2 cos UdU ! dx 2 cos U ! 2 cos UdU ! dU 2 cos U x 1 I ! U k ! arcsen 2 I ! 4 (1 x) 2

MATEMTICA II

9.- SOL:

xdx 4 x22tgU ! x 2tgU secUdU ! dx 2 secU ! 4 x 2

I !

2tgU .2tgU . sec UdU d ! 2 tg 2UdU ! 2 sec 2 U U 1 2 secU

I ! 2 dU 2 sec 2 UdU ! 2U 2tg U k I ! 2 arctg x

2 x k x 1 4 x2 dx

10.- SOL:

2 sen U ! x 2 cos UdU ! dx 2 cos U ! 4 x 2

I !

2 senU 12 cos UdU2 cos U

!

2 senU 1dU ! 2 senUdU dU

I ! 2 cos U U k ! 4 x 2 arcsen x

2

11.-

dx 2 5x 269

UNAC-FIEE SOL:2 senU ! x 5 2 cos UdU ! dx 5

MATEMTICA II

2 cos U ! 2 5 x 2

2 cosUdU 1 ! dU 5. 2 cosU 5 1 1 I! U k ! arcsen 10 x 2 k 5 5 I!

12.- SOL:

senxdx 2 cos2 x2 sen U ! cos x 2 cos UdU ! senxdx 2 cos U ! 2 cos 2 x

I !

2 cos UdU ! dU 2 cos U

2 cos 2 x k I ! U k ! arccos 2

13.- x 2 16 x 2 dx SOL:4 senU ! x 4 cos UdU ! dx 4 cos U ! 16 x 2

70

UNAC-FIEE I ! 16sen 2U .4 cosU .4 cosUdU ! 64 4 sen 2U cos2 UdU ! 64 sen 2 2U dU 1 cos 2 2U

MATEMTICA II

sen4U I ! 64 dU 64 cos2 2UdU ! 32 cos 4UdU ! 32 k 4 2 x 16 x I ! 32 4 4

16 x 2 4

x 2 K ! x 16 x 2 4 2

8 4x

2

K

14.- x 2 3 x 2 dx SOL: 3senU ! x 3 cosUdU ! dx 3 cosU ! 3 x 2

I ! I !

3 sen U .2 2

3 cos U . 3 cos UdU ! 9 sen 2U cos 2 UdU !

9 2 sen 22U dU 4 1 cos 2U

9 sen 4U 9 9 9 2 dU 4 cos 2UdU ! 8 cos 4UdU ! 8 4 k 4

5 senU ! x 5 cos UdU ! dx 5 cos U ! 25 x 2

1 cscUdU 5 2 1 1 I ! ln ctgU cscU k ! ln 25 x 5 k x x 5 5 dx 16.- x2 4 x2 SOL: I !

5senU .5 cosUdU

5 cos UdU

!

71

9 x 3 x 2 I ! 8 3 3 dx 15.- x 25 x 2 SOL:

3 x 2 x 2 1 K ! x 3 x 2 3 2 x 2 3 3 8

UNAC-FIEE2 sen U ! x 2 cos UdU ! dx 2 cos U ! 4 x 2 2 cos UdU2

MATEMTICA II

I !

4 sen U 2 cos U

!

1 1 2 csc UdU ! 4ctgU k 4

1 4 x2 I ! 4 x

k

17.- SOL:

dx 4x x2 ctgU ! 2 x1 2

cscUdU ! cscU !

2dx x

4x x2 2 x

I !

1 x csc 2 UdU 2 x csc U ! 2 csc UdU 4x x2 k 2 x

1 1 1 I ! ln ctgU cscU k ! ln 2 x 2 2 2

18.- SOL:

dx

4x

2

9 2 3 csc U ! 2 x 3 csc UctgUdU ! 2 dx 3ctgU ! 4x2 9

3

I!

1 1 cscUdU 3 cscUctgUdU ! senU cos 2 U dU dt ! 2 cos 1 UsenUdU ! 3 2 18 18 ctg U 23ctgU t

72

UNAC-FIEE1 3 1 t 2 dt 1 2t 2 I ! ! k 18 2 36 3

MATEMTICA II

cos 3 U 4 x3 k ! 2 54 27 4 x 9 dx 19).- 5 4 x x2 SOL: I !

K 3

3senU ! x 2 3 cosUdU ! dx 3 cosU ! 9 x 2 I!2

27 cos U3

3 cosUdU

1 I! 9 20).- SOL:

1 1 2 sec UdU ! 9 tgU k 9 x2 K 2 9 x 2 ! dx

x a2 x2actgU ! x a csc 2 UdU ! dx a cscU ! a 2 x 2

a csc 2 UdU 1 1 I ! ! sec UdU ! ln sec U tg U k actg Ua csc U a a 1 a2 x2 a K I ! ln a x x

21).- SOL:

dx x x2 a 2a cscU ! x a csc UctgUdU ! dx actg U ! x2 a2

73

UNAC-FIEE a cscUctgUdU 1 U ! dU ! a cscUactgU a a 1 K I ! x a arcctg a

MATEMTICA II

I!

22).- SOL:

dx x x2 a 2asenU ! x a cos UdU ! dx a cos U ! a2 x2

I! I!

asenUa cosU

a cosUdU

!

1 1 cscUdU ! a ln ctgU cscU k a

1 a2 x2 a ln K a x x dx

23).- SOL:

a

2

x2

3

asenU ! x a cos UdU ! dx a cos U ! a2 x2

I !

1 1 a cos UdU ! 2 sec 2 UdU ! 2 tgU k 3 3 cos U a a x 1 K I ! 2 2 a a x2

a

24).- SOL:

dx

a

2

x2

2

74

UNAC-FIEEactgU ! x a csc 2 UdU ! dx a cscU ! a 2 x 2 1 1 1 a csc 2 UdU 2 I ! 4 ! 3 sen U ! 3 dU 3 cos 2 UdU Ud 4 a csc U a a a 1 cos 2 U arcctg 1 U sen 2U U I ! 3 3 k ! 4 a a 2 a3 a3

MATEMTICA II

ax x a 1 arcctg x a a 2

2

x2 K 2

25).- SOL:

e x dx

4e

2x

1

3

tgU ! 2e x sec 2 UdU ! 2e x dx secU ! 4e 2 x 1

I !

sec 2 UdU 1 1 2 sec 3 U ! 2 cosUdU ! 2 senU k 1 2e x K I ! 2 4e 2 x 1

26).- SOL:

csc2 xdx

4 ctg x 2 3

2senU ! ctgx 2 cosUdU ! csc 2 xdx 2 cosU ! 4 ctg 2 x

I !

2 cosU

2 cos UdU3

!

1 1 2 sec UdU ! 4 tgU k 4

1 ctgx K I ! 4 4 ctg 2 x 75

UNAC-FIEE

MATEMTICA II

27).-

x

2

x 1 dx

SOL: 3 tgU ! x 1 2 2 3 sec 2 UdU ! dx 2 3 secU ! x 2 x 1 2 3 9 3 sec 3 UdU ! sec5 U " A" por partes : 16 2 A

I !

3 sec 2 U 2 u dv

3 2 A ! sec secUdU du ! 3 sec 2 U sec UtgUdU v ! tgU U

A ! sec UtgU 3 tg 2U sec 3 UdU ! sec 3 UtgU 3 sec5 UdU 3 sec3 UdU ] 23 sec U 1 A

sec UtgU ln secU tgU c 4 A ! sec 3 UtgU 3 2 2 3 sec Utg U 3 secUtgU 3 ln sec U tgU A! 4 8 8 9 sec3 UtgU 27 secUtgU 27 ln sec U tgU @I ! k 64 128 128

x @I !SOL:

2

x 1 2 x 1 9 x 2 x 1 2 x 1 8 64

3

28).- x 2 x 1dx

3 tgU ! x 1 2 2 3 sec 2 UdU ! dx 2 3 secU ! x 2 x 1 2

76

27 ln

2 x2 x 1 2x 1 3 128

K

UNAC-FIEE 3 9 3 sec 3 UdU ! sec 5 U " A" por partes : sec 2 U 2 16 2 Au dv 3

MATEMTICA II

I !

3 2 A ! sec secUdU du ! 3 sec 2 U sec UtgUdU v ! tgU U

A ! sec Utg U 3 tg 2U sec 3 UdU ! sec 3 UtgU 3 sec 5 UdU 3 sec 3 UdU ] 23 sec U 1 A

secUtgU ln sec U tg U c 4 A ! sec3 UtgU 3 2 2 sec 3 UtgU 3 sec UtgU 3 ln secU tgU A! 4 8 8 9 sec3 UtgU 27 secUtgU 27 ln sec U tgU @I ! k 64 128 128

@I !

x

2

x 1 2 x 1 9 x 2 x 1 2 x 1 8 64

3

29).-

dx

x

4

1

x4 1 x2

SOL: tgU ! x 2 sec 2 UdU ! 2 xdx secU ! x 4 1 sec 2 UdU secU tgU 1 cosUdU 2 1 senU senU dU senU ! t @ cosUdU ! 2tdt 2 tgE ! t sec2 EdE ! dt I! I! 1 sec 2 EdE 1 1 secE ! 2 secEdE ! 2 ln secE tgE 2 t 2 1 ! secE

I! I!

2 U sec2

tgU

!

1 2tdt 1 dt t 2 1t ! 2 t 2 1 2

1 1 1 ln t 2 1 t c ! ln senU 1 senU k ! ln 2 2 2

77

27 ln

2 x2 x 1 2 x 1 3 128

K

x2 x4 1

1

x4

x4 1

K

UNAC-FIEE

MATEMTICA II

30).- SOL:

x 3 1 x 4 dx 1 x4 x4 1 x4

sen U ! x 2 cos UdU ! 2 xdx 1 x 4 ! cos U

I! I! I!

2sen1 2 1 2

sen 2U cosU cosUdU1 2

3

U cosU sen 2U cos 2 U senUdU cosU ! u @ senUdU ! du cosU sen 2U du du 1 1 ! ! ln u 1 2 2 2 2 2 u 1 u u 1 5 2 4

!

senU cos2 UdU 1 2 cosU senU cos 2 U

2

12 5 4 C u

1 I ! ln cosU 1 2 2

U 12 5 4 k cos2

1 I ! ln 1 x 4 1 2 2

1 x

4

1

54 K 22

78

UNAC-FIEE PRACTICA N 6 A) CALCULAR LAS SIGUIENTES INTEGRALES RACIONALES: 1.-

MATEMTICA II

x 1x 5x 3SOLUCIN:

dx

x 1x 5x 3 ! x 1 x 5 x 31 ! A B C x 2 2 A 2 B 6C x 15 A 3B 5C 1 A! A BC ! 0 16 1 2 A 2 B 6C ! 0 B! 32 1 C! 15 A 3 B 5C ! J 32 LUEGO: dx 1 dx 1 dx 1 dx ! ! x 5 32 x 3 x 1x 5x 3 16 x 1 32 1 1 dx 1 ! ! ln x 1 ln x 5 ln x 3 x 1x 5x 3 16 32 32 2.-

dx

A

B

C

2 x 12 x 3SOLUCIN:

x 2 dx

2 x 12 x 3 ! 2 x 1 2 x 3 2 x 12 x 3 ! 2 x 1 2 x 35 A! 2 A 2B ! 1 4 3 3A B ! 2 B! 4 LUEGO: 5 dx 3 dx x 2 dx 2 x 12 x 3 ! 4 2 x 1 4 2 x 3 3 5 x2 2 x 12 x 3 ! 8 ln 2 x 1 8 ln 2 x 3 x2 A B

x 2 dx

Adx

Bdx

79

UNAC-FIEE

MATEMTICA II

3.-

x 2 x 2 SOLUCIN:

x 3dx

x 2 x

x 3 ! A B x 2 A

Adx Bdx 2 x 2 x 2 x3 A B ! x 2 x 2 x 2 x 2

x 3dx

!

2B

3 2 1 3 2 , B! A A! B !1 2 2 2 2 2 A 2B ! 3 LUEGO: x 3dx 3 2 dx 3 2 1 dx x 2 x 2 ! 2 2 x 2 2 2 x 2

x 4.-

x 3dx2 x 2

!

3 2 1 3 2 ln x 2 ln x 2 2 2 2 2

x 2x 12 x 1SOLUCIN:

x

2

9 dx

x 2x 12 x 1 ! x 2 x 1 2 x 1x2 9 A B C x 2x 12 x 1 ! x 2 x 1 2 x 1 x 2 9 ! 2 A 2 B C x 2 A 5 B C x 2 B A 2C 19 A! 2 A 2B C ! 1 9 31 10 , c! A 5B C ! 0 B! 9 9 2 B A 2C ! 9 LUEGO: 19 dx 10 dx 31 dx x 2 9 dx x 2x 12 x 1 ! 9 x 2 9 x 1 9 2 x 1 31 10 19 x 2 9 dx x 2x 12 x 1 ! 9 ln x 2 9 ln x 1 18 ln 2 x 1

x

2

9 dx

Adx

Bdx

Cdx

80

UNAC-FIEE

MATEMTICA II

5.-

3x

x2 1 dx x3 x2 2x2

SOLUCIN:

x

2 x3

1 A B C dx dx ! x 2x xx 2x 12 2

2x2 1 A B C ! 3 2 x x 2x x x 2 x 1 2 x 2 1 ! A B C x 2 B A 2C x 2 A 1 A! A B C ! 2 2 3 B A 2C ! 0 B! 2 2A ! 1 C !1 2x 2 1 1 dx 3 dx dx x3 x 2 2 x ! 2 x 2 x 2 x 1 3 2x 2 1 1 x 3 x 2 2 x ! 2 ln x 2 ln x 2 ln x 1 6.5 x2 x 3 x dx SOLUCIN:

A B C dx 5 x2 x 3 x dx ! xx 1x 1 5 x2 A B C ! 3 x x x x 1 x 1 x 2 5 ! A B C x 2 C B x AA B C !1 CB!0 A!5 A ! 5 B!3 C !3

5 x dx dx dx 3 dx ! 5 3 3 x x x 1 x 1 2 5 x x 3 x dx ! 5 ln x 3 ln x 1 3 ln x 1

x

2

81

UNAC-FIEE

MATEMTICA II

7.-

4x

2 x 5 dx3

x

SOLUCIN:

4x

2 x 5dx ! A B C dx 3 x2 x 12 x 1 x

B C 2x 5 A ! 3 4x x x 2x 1 2x 1 4 x 5 ! 4 A 2 B 2C x 2 C B x A A 4 !5 A 2 B 2C ! 0 B ! 7 CB!4 ! 3 A ! 5 C

4x

dx dx dx 7 3 x x 2x 1 2x 1 2 x 5dx ! 5 ln x 7 ln 2 x 1 3 ln 2 x 1 4x3 x 2 23

2 x 5dx ! 5

8.-

x

3

3 dx 2x x 22 2

x

SOLUCIN: FALTA 3x 2 2 x 1 9.- 3 dx 6 x 7 x 2 3x

SOLUCIN:

A B C dx 2x 1 dx ! 3 2 7 x 3x 6 x 7 x 2 3x 3x 2 2 x 1 A B C ! 3 2 6 x 7 x 3x x 3 x 1 2 x 3 3 x 2 2 x 1 ! 6 A 2 B 3C x 2 C 7 A 3 B x 3 A 1 A! 6 A 2 B 3C ! 3 3 6 B! C 7 A 3 B ! 2 11 43 C! 3A ! 1 33

6x

3x3

2

6x

3x3

2

2x 1 1 dx 6 43 dx dx dx ! 2 7 x 3x 3 x 11 3 x 1 33 2 x 3

82

UNAC-FIEE

MATEMTICA II

6x10.-

3x3

2

43 2 1 2x 1 ln 2 x 3 dx ! ln x ln 3 x 1 2 66 11 3 7 x 3x

x 3 3x 2 x 4 3x 2 2 dx SOLUCIN:

A B C D dx x3 3x 2 x 4 3x 2 2 dx ! x 1x 1 x 2 x 2

x 3x 2 A B C D ! 4 2 x 3x 2 x 1 x 1 x 2 x 2 x 3 3 x 2 ! A B C x 3 B A 2C 2 D x 2 2 A 2 B C D x3

2 A 2 B 2C 2 D A B!0 B C D !1 B A!2 A 2C 2 D ! 0 2 2 2 A 2 B C D ! 3 C! 2 2 22 A 2 B 2C 2 D ! 2 D! 2 2 2 LUEGO: dx 2 2 dx 22 dx x 3 3x 2 x 4 3x 2 2 dx ! 2 x 1 2 2 x 2 2 2 x 2 22 2 2 x 3 3x 2 x 4 3x 2 2 dx ! 2 ln x 1 2 2 ln x 2 2 2 ln x 2

11.-

x 1 x 1x2 5 x 2 5 dx3

x

2

5 dx3

SOLUCIN:!

A B C dx x 13

x 1

3

!

A B C 2 x 1 x 1 x 13 A !1 B!2 C!6

x 2 5 ! Ax 2 B 2 Ax A B C A !1 B 2 A ! 0 BC !5 A

83

UNAC-FIEEx2 5 dx dx dx 2 6 2 x 1 x 1 x 13 2 3 x 1 x 12

MATEMTICA II

x 1 x 112.x2 5

3

dx !

3

dx ! ln x 1

x 3x 2 x 2 3

x

3

4 x 5 dx

SOLUCIN:

x 3x 2 x ! 2 3

x

3

4 x 5 dx

A B C D E F dx x 3x 2 2 x 3A B C2

x 3x 2 x ! x 3 x 2 x 22 3

x3 4x 5

D E F x x 2 x3

x 3 4 x 5 ! A B C x 5 4 A B C D E x 4 4 A 6 B 3C 8 D E F x 3 12 D 8 E F x 2 12 E 8 F x 12 F 34 191 A D! A! BD!0 675 36 14461 11 A B C D E ! 0 B! E! 4 2700 18 21 5 4 A 6 B 3C 8 D E F ! 1 C! F! 40 12 12 D 8 E F ! 0 12 E 8 F ! 4 12 F ! 5 x 3 4 x 5 dx 34 dx 14461 dx 21 dx 191 dx x 3x 22 x 3 ! 675 x 3 2700 x 2 40 x 22 36 x

11 dx 5 dx 18 x 2 12 x 3 34 14461 21 191 x 3 4 x 5 dx x 3x 22 x 3 ! 675 ln x 3 2700 ln x 2 40x 2 36 ln x

13.-

x

11 5 18x 24 x 2

3x

2

x 2 dx 3

x 3 2 2

SOLUCIN:

84

UNAC-FIEE

MATEMTICA II

x 3 x 3 ! x 3 x 3 2 2 2 2

3x

2

x 2 dx

A B C D dx

3x 2 x 22 2

x 3 x 3

!

A B x 3 x 3

3 x 2 x 2 ! A C x 3 3 A B 3C D x 2 3 A 2 3 B 3C 2 3 D x 3 3 A 3 B 3 3C 3 D A C ! 0 3 A B 3C D ! 3 3 A 2 3 B 3C 2 3 D ! 1 3 A 3B 3 3C 3 D ! 2 3 A! 7 3 11 3 ; B! 36 12 21 3 11 3 ; D! C! 12 12

2

C D x 3 x 3

2

x 3 x 3 ! 2 2

3x

2

x 2 dx

7 3 dx dx 1 3 x 3 12 x 3 36

2

dx 21 3 x 3 12

dx 11 3 x 3 12

2

!

7 3 11 3 21 3 11 3 ln x 3 ln x 3 36 12 12 x 3 12 x 3

15.-

x xx x

2

3 / 4 dx 2x 1

2

3

SOLUCIN:2

dx 3 / 4 2 x 16 3

2

!

x

3 / 4 A B C D E F dx dx ! 6 x 1 x 162

x2 3/ 4

x 1

!

A B C D E F 2 3 4 5 x 1 x 1 x 1 x 1 x 1 x 16

x2

3 ! Ax 5 5 A B x 4 A 4 B C x 3 A 6 B 3C D x 2 10 10 4 5 A 4 B 3C 2 D E x A B C D E F

85

UNAC-FIEEA!0 A!0 A B ! 0 B!0 5 A 4B C ! 0 C!0 10 10 A 6 B 3C D ! 1 D !1 E ! 2 5 A 4 B 3C 2 D E ! 0 3 1 A F! BC DE F ! 4 4 2 x 3/ 4 dx dx 1 dx dx ! 2 6 4 5 x 1 x 1 4 x 16 x 1

MATEMTICA II

x 116.-

x

3x 5dx x 2x x 2 x 4 3 2

3x 5dx 3x x 2x x 2 x ! x 2xx4 3 2

RESOLVIENDO I: A B Cx dx 3x 3 -! ! 2 x 2 x x 2 x 2 x 2 x 2 3x 3 A B Cx ! 2 2 x 2 x x 2 x 2 x x 2 3 x 3 ! A C x 2 A B 2C x 2 2 B A A ! 1 C ! 0 A B 2C ! 0 B !1 2B ! 0 2 C !1

3x 3 dx dx x 1 ! 2 2 x2 x x2 x 2 x x 2 RESOLVIENDO II: 5 dx D E Fx H dx -- ! ! 2 x 2x x x 2 x 2 x 2 x 2 x 5 dx D H E Fx ! 2 2 x 2 x x x 2 x 2 x x x 2 5 ! D H F x 3 D 3H E 2 F x 2 2 D 4 H 2 E x 4 H -!

2

3/46

dx !

3

x 1

3

1 2x 14

1 20x 15

SOLUCIN:4

2

5dx x 2 x x 2 x 2 x2 3x3 5dx ! 2 x 2 x x 2 x 2 x x 2 x 2

-

--

86

UNAC-FIEE5 8 5 4 , E! 15 8

MATEMTICA II

D H F !0 D 3H E 2 F ! 0 2 D 4H 2E ! 0 4H ! 5

D!

5 dx 5 dx 5 dx 5 x 3dx ! F 2 x 2x x x 2 8 x 2 4 x 8 x 2 x 2 SUMANDO "E F " -- !

@

3x 5dx ! 3 dx 5 dx 5 x 3dx x 1dx x 2x x 2 x 8 x 2 4 x 8 x x 2 x x 24 3 2 2 2

3 dx 5 dx 5 2 x 1 dx ! 2 dx 5 2 8 x 2 4 x 16 x x 2 1 7 x 2 4 dx 1 2 x 1 dx 3 2 2 2 x x2 1 7 x 2 4

!

3 dx 5 dx 13 2 x 1dx 1 dx x 2 4 x 16 x 2 x 2 16 1 2 7 8 x 2 4 3 x 4 5 dx 3 5 13 1 2x 1 2 x 2 x 3 x 2 2 x ! 8 ln x 2 4 ln x 16 ln x x 2 8. 7 arc tg 7 c

17.-

xx

x6 3

3

4x 1 dx 2x4 x2 SOLUCIN:

x6

4x 1 A B C D E F dx dx ! 4 2 2 2 2x x x 2 x 1 x 1

x3 4 x 1 A B C D E F ! 2 6 4 2 2 x x x 1 x 1 x 1 x 12 x 2x x x 3 4 x 1 ! A C E x 5 B C D F E x 4 2 A C 2 D E 2 F x 3 2 B C D E F x 2 Ax B87

!

H ! F! 5 8

UNAC-FIEEAC E ! 0 C D F E ! 0 B 2 A C 2 D E 2F ! 1 2B C D E F ! 0 A !4 B ! 1 A!4 B ! 1 C ! 5 / 4 D !1 E 11 / 4 F ! 3 / 2

MATEMTICA II

x18.-

x

5 x 3 3dx ! 5 x 3 3dx 3 dx ! x 4 8 x 2 16 x 2 4 2 x 2 2 x 2 23

x 2 x 2 ! x 2 x 2 2 2 2

x 2 x 2 5 x 3 3 ! A C x 3 2 A B 2C D x 2 2 A 4 B 4C 4 D x 4B A 2C D 2 2

A A ! 83 / 24 C ! 5 A B 2C D ! 0 2 B ! 37 / 16 2 A 4 B 4C 4 D ! 0 C ! 37 / 24 A 2C D ! 3 B D ! 43 / 16 4 37 43 5x3 3 83 dx 37 dx dx dx x 4 8 x 2 16 dx ! 24 x 2 16 x 2 2 24 x 2 16 x 22

x19.-

4

2x

"6 4

11 dx 3 4x 1 dx dx 5 dx dx dx dx ! 4 1 2 4 2 2 4 x 1 x x x 2x x x 1 4 x 1 2 x 123 6

x

x

3

4x 1 1 5 1 11 3 dx ! 4 ln x ln x 1 ln x 1 c 4 2 2x x x 4 2x 1 x 1 43

5x5 x

3 dx 8 x 2 16 SOLUCIN:

5 x

3

3 dx

A B C D dx2

5x 3 3

!

A B C D 2 x 2 x 2 x 2 x 2 2

5 x

43 37 37 3 83 ln x 2 dx ! ln x 2 2 16x 2 16x 2 24 8 x 16 243

5

3 x 4 dx x 16 x 3 8 x 2 32 x 162 4

8x

SOLUCIN:

88

UNAC-FIEE 3 x 4 dx 8 x 2 3 x 4 dx ! 2 x 5 x 4 16 x 3 8 x 2 32 x 16 x 2 2 x 2 2 2 x 12

MATEMTICA II

8 x

! 8 x 2 3x 42 2

x 2 x 2 2 x 12 2

A B C D E dx

x 2 x 2 2 x 1 8 x 2 3 x 4 ! 2 A E 2C x 4 3 A 2 B 5C 2 D x 3 6 A 7 B 2C 9 D 4 E x 2 6 A 2 B 10C 12 D x 4A B C D E 2 A E 2C ! 0 A ! 7 /8 3 A 2 B 5C 2 D ! 0 B ! 7 / 8 6 A 7 B 2C 9 D 4 E ! 8 C ! 11 / 120 A 2 B 10C 12 D ! 3 6 D ! 3/8 A B C D E ! 4 4 E ! 8 / 15 8 x 2 3 x 4 dx 7 dx 7 dx dx 11 2 x 5 x 4 16 x 3 8 x 2 32 x 16 ! 8 x 8 8 x 22 120 x 2

!

A B C D E 2 2 x 2 x 2 x 2 x 2 2 x 1

!

7 7 11 ln x 8 ln x 2 8 8x 2 120 3 8 ln 2 x 1 8x 2 15

20.-

x

3 x 2 2 dx 2 dx ! ! x 4 x3 3x 2 5x 2 x 2 x 132

3x 2 23

x 2 x 1 3 x 2 2 ! A B x 3 3 A C x 2 3 A 3B C D x A 2 B 2C 2 DA B ! 0 A C ! 2 3 3 A 3B C D ! 0 2 B 2C 2 D ! 2 A A ! 5/9 B ! 5 / 9 C ! 1/ 3 D ! 5 / 389

#4

8 dx dx 3 x 22 15 2 x 1 8

2 dx x 3x 2 5 x 22 3

3x

SOLUCIN:

3x

x 2x 1

A B C D dx3

!

A B C D 2 x 2 x 1 x 1 x 13

$

UNAC-FIEE

MATEMTICA II

5 5 dx 5 dx 1 dx dx 2 dx ! 2 2 x x 3 x 5 x 2 9 x 2 9 x 1 3 x 1 3 x 132 4 3

3 x

5 5 1 5 5 ! ln x 2 ln x 1 2 9 9 3x 1 6x 1 6x 12 21.-

x x

5dx 2 3

SOLUCIN:dx 5dx ! 5 2 2 3 x 3 x 7 dx 2 x 5 x2 2 ! 5 x arc tg c 3 3

2

22.-

x

SOLUCIN:

x 7 dx2

x 2 5 x2 2 5 x 2 x7 A Bx C Dx ! 2 2 2 x 5 x 2 x 5 x2 2 x 7 ! B D x 3 A C x 2 2 B 5 D x 2 A 5C BD!0 A ! 7/3 C !0 A B ! 1 / 3 2 C ! 7 / 3 B 5D ! 1 A 5C ! 7 2 D ! 1/ 3 x 7 dx 1 7 x dx 1 7 x dx x2 5 x2 2 ! 3 x2 5 3 x2 22

!

A Bx C Dx

dx

1 7 7 x 1 x ln 2 2 ! 5 arc tg 3 2 ln x 5 2 arc tg 2 2 x 2 c 3

23.-

x

A Bx C Dx

dx 4 x 3 4 dx ! dx ! 2 2 x4 4x2 3 x 3 x 1 x 2 3 x2 13

%4

1 7 dx xdx 7 dx xdx ! 2 2 2 2 3 x 5 x 5 x 2 x 2

x x

4 dx 4x2 33

SOLUCIN:

90

UNAC-FIEE x3 4 A Bx C Dx 2 ! 2 2 2 x 3 x 1 x 3 x 1 3 3 x 4 ! B D x A C x 2 B 3 D x A 3C B D !1 A ! 2 C ! 0 A B ! 3/ 2 B C !2 3D ! 0 3C ! 4 A D ! 1 / 2

MATEMTICA II

1 4 1 x 3 2 2 ! arc tg ln x 3 4 arc tg x ln x 1 2 2 3 3 2

24.-

x 3x 5dx 3x 1x 33 2 2

x 3x 5dx A Bx C Dx

dx 3x 1x 3! 3x 1x 33 2 2 2 2

x 3 3x 5 A Bx C Dx ! 2 2 2 3x 1 x 3 3 x 1 x 2 3 x 3 3 x 5 ! B 3 D x 3 A 3C x 2 3 B D x 3 A C B A ! 15 / 8 3D ! 1 3C ! 0 A B ! 5 / 4 3 C ! 5 / 8 B D ! 3 A C ! 5 3 D ! 3/ 4

x 3x 5dx ! 1 15 10 x dx 1 5 6 x 8 x 3 3x 1x 3 8 3x 13 2 2 2 2

25.-

x

&4

4 1 4 3x 1 4 x dx ! 2 2 2 x 3 2 x2 1 x 4x 3 1 dx xdx dx xdx ! 4 2 3 2 4 2 2 2 x 3 x 3 x 1 x 1 3 4

x

SOLUCIN:

1 15dx xdx dx xdx ! 2 10 2 5 2 6 2 8 3x 1 3x 1 x 3 x 3 1 5 5 x 2 arc tg ! 15arc tg 3 x ln 3 x 2 1 3 ln x 3 c 8 3 3 3

x

3 dx 5x2 63

SOLUCIN:91

UNAC-FIEE

MATEMTICA II

A Bx C Dx

dx 3 x 3 3 dx ! dx ! 2 2 x4 5x2 6 x 3 x 2 x2 3 x2 23

x

3 A Bx C Dx ! 2 2 x 3 x 2 x 3 x2 2 x 3 3 ! B D x 3 A C x 2 2 B 3 D x 2 A 3C B D !1 C ! 0 A 2 B 3D ! 0 A 3C ! 3 2 3

x

2

x

3 3 x dx 3 2 x dx 3 dx ! 2 2 x2 2 5x 6 x 3 dx xdx dx 2 xdx ! 3 2 3 2 3 2 2 2 x 3 x 3 x 2 x 2 3 x 3 3 3 x 3 x 2 2 x 4 5 x 2 6 dx ! 3 arc tg 3 2 ln x 3 2 arc tg 2 ln x 2 c 3 4

26.-

x x

x 82

LA INTEGRAL I. LO DESARROLLAMOS POR EL MTODO DE REDUCCIN dx x 3x 3 x arc tg - ! 8 ! 8 3 2 28 x 2 7 2 392 x 2 7 392 7 7 x 7

27.-

x x

3x2 5x 24

'

x

x 82

7

3

dx

SOLUCIN:dx ! xdx 8 xdx

7

3

x

2

7

3

x

2

7

3

2

x 8

x4

7

3

dx !

3x 3 x x 8 28 x 2 7 392 x 2 7 392 7 arc tg 7 4 x2 7 1

2

3x 2 5 x 2 4 x 2 32

dx

SOLUCIN: A Bx C Dx E Fx G Hx dx 2 dx ! 2 2 2 x 1 x 3 4 x2 3 x2 1 x2 3

2

92

UNAC-FIEE3x 2 5x 2 A Bx C Dx E Fx G Hx 2 2 2 x 2 1 2 1 x 3 2 3 x x

MATEMTICA II

x

4

4 x 32

2

!

3 x 2 5 x 2 ! F B x 7 A E x 6 5 F 7 B H D x 5 7 A 5 E G C x 4 7 F 15 B 2 H 9 D x 3 A 7 E 2G 9C x 2 3 F B H 9 D x 9 A 3 E G 9C 15 F A ! 3 / 2 B!0 E!0 A B ! 31 / 12 F 7 B H D ! 0 5 C !1 7 D ! 2/3 A 5E G C ! 0 7 E ! 3/ 2 F 15 B 2 H 9 D ! 0 A 7 E 2G 9C ! 3 15 F ! 31 / 12 3 G!2 F B H 9 D ! 5 A 3 E G 9C ! 2 9 H ! 35 / 6 3x 2 5x 2 1 18 31x 1 3 2 x 1 18 31x 1 12 35 x x 4 4 x 2 32 dx ! 12 x 2 1 3 x 2 12 12 x 2 3 6 x 2 32

x

3x2 5 x 24

4x 32

2

dx !

3 31 3 x 31 ln x 2 1 ln x 2 1 arc tgx arc tg 2 4 24 3 24

!

1 4 1 x2 2

6 x 3

x arc tgx c 8 x 1 2

28.-

x x

6 x 5 dx2

1

3

SOLUCIN:

6 x 5 dx !2

x @

1 6 x dx2

3

x

6 x dx2

5.2 2 x2 1 x 3x 3 ! arc tgx 3 2 2 8 x 1 8 x2 1 4 x2 1 1 dx

3

!

1 3

3

5

dx

x

2

1

3

x

6 x 5 dx ! 2

3 2 2 1 x2

1

3

3x 3 x 2 arc tgx c 2 8 1 8 x 1 x

29.-

2 x

x 9 dx2

1

SOLUCIN:93

UNAC-FIEE

MATEMTICA II

2x

x 9 dx !2

1

2x

xdx dx 9 2 2 1 2x 1 1 9 ln 2 x 2 1 arc tg 4 2

2x

30.-

x x 4 x 5 dx 32 2 2

SOLUCIN:

x x 4 Ax B Cx D

dx 3x 5 dx ! 3x 52 2 2 2 2

x2 x 4

3 x2

2

5

2

!

Ax B Cx D 2 2 3x 5 3x 2 5

2

x x 4 ! 3 Ax 3Bx 5 A C x 5 B D 3A ! 0 A!0 B ! 1 3 B ! 1/ 3 5 C ! 1 A C ! 1 B D ! 4 5 D ! 7/3 x 2 x 4 dx ! 1 dx 1 7 x ! 1 dx 1 xdx 2 2 3 x 2 5 3 x 2 5 3 x 2 5 3 x 2 5 3 3 3 3 3x 2 523

3x 3 3 1 3x 1 dx xdx -! ! ! arc tg 2 5 9 2 5 2 9 2 5 5 5 3x 2 5 10 x x 3 3 2 3x x x4 22 1 dx ! arc tg 3x 2 5 2 5 90 x 2 5 5 21x c 3 15

31.-

A Bx C

x2 3 3 x 3 5 x dx ! x x 2 5 dx ! x x 2 5 dx x 2 3 A Bx C ! 2 x3 5x x x 52

(

7 xdx 3x 2 52 3

x3

3 dx x 5x2

SOLUCIN:

x

94

UNAC-FIEE x 2 3 ! A B x 2 Cx 5 A A A ! 3/ 5 B ! 1 C ! 0 B ! 8 / 5 A ! 3 5 C!0 x2 3 3 dx 8 xdx 1 2 x 3 5 x dx ! 5 x 5 x 2 5 ! 5 3ln x 4 ln x 5 c

MATEMTICA II

32.-

x x

x4 2

3 dx 5x2 42

SOLUCIN:

Ax B Cx D

dx 3 dx ! 2 4 x 1 x2 4 x2 1 x2 3 Ax B Cx D 2 ! 2 2 2 x 4 x 1 x 4 x 1 2 3 x 3 ! A C x B D x 2 A 4C x B 4 D A A!0 C ! 0 D !1 B B!5 A C!0 4C ! 0 4D ! 3 B D ! 2 / 3 2

x

3 dx dx 2 dx ! 5 2 2 2 x 4 x 1 x 4 3 x 1 5 x 2 ! arc tg arc tgx c 2 2 32 2

x

33.-

2 x

5x 3 dx x 272 3

SOLUCIN:

2 x

5x 3 2 x 2 5 x 3 dx dx ! x 3 x 2 3 x 9 x 3 272

2 x 5x 3 A Bx C ! 2 2 x 3 x 3x 9 x 3 x 3x 9 2 x 2 5 x 3 ! A B x 2 3 B C 3 Ax 9 A 3C2

95

UNAC-FIEE

MATEMTICA II

A A!0 B!2 B!2 3 B C 3 A ! 5 A 3C ! 3 9 C !1 2 x 3dx 2 x 2 5x 3 2x 1 4dx ! 2 dx ! 2 3 x 2 3x 9 x 3x 9 x 3x 9 x 27 8 2x 3 arc tg ! ln x 2 3 x 9 c 3 3 3 3

34.-

x

x5

3x 1 dx 5x 3 4 x3

SOLUCIN: 3x 1 dx ! 5 x 5x 3 4 x3 3

x

x xx

3 2

3 x 1 dx 4 x2 1

x 3x 1 A Bx C Dx E 2 ! 2 2 2 x x 4 x 1 x x 4 x 1 3 4 x 3 x 1 ! A B D x C E x 3 5 A B 4 D x 2 C 4 E x 4 A A A ! 1/ 4 BD !0 E !1 C B ! 1 / 12 5 A B 4 D ! 0 C ! 7/3 4 E ! 3 C D ! 1 / 3 4 E ! 4 / 3 A !1

x 3x 1dx 1 dx 1 x 28 1 x 4 xx 4x 1! 4 x 12 x 4 dx 3 x 1 dx x 3x 1 dx ! 1 x 1 x 4 14arc tg x 1 lnx3 2 2 2 2

x

3

2

2

5

5x 4 x3 2

4

24

2

6

1

4 arc tgx c 3

35.-

x

x3 2

3x 5 dx x2 4x 6 SOLUCIN:

3x 5 x 2 3 x 5 dx dx ! x 3 x 2 4 x 6 x 3 x 2 2 x 2 x 2 3x 5 A Bx C ! 2 2 x 3 x 2 x 2 x 3 x 2 x 2 2 x 3 x 5 ! A B x 2 2 A 3 B C x A 3C

x

96

UNAC-FIEE

MATEMTICA II

A A ! 5 / 16 B !1 2 A 3 B C ! 3 B ! 11 / 16 3C ! 5 A C ! 25 / 16 5 dx 1 11x 25 x 2 3x 5 x 3 x 2 4 x 6 dx ! 16 x 3 16 x 2 2 x 2 11 1 ! 5 ln x 3 ln x 2 2 x 2 36arc tg x 1 2 16

36.-

x

x4

4x 3 dx x2 2x 23

SOLUCIN: 4x 3 x 2 x 3 dx dx ! x4 x2 2x 2 x 1 x 2 2 x 23

x

x x3 A Bx C ! 2 2 x 1 x 2 x 2 x 1 x 2 x 2 x 2 x 3 ! A B x 2 2 A C B x 2 A C2

A A ! 1 / 5 B !1 B ! 6/5 2 A C B ! 1 A C ! 3 2 C ! 13 / 5 x3 4x 3 1 dx 1 6 x 13 4 dx dx ! 2 x x 2x 2 5 x 1 5 x 2 2x 2 1 ! ln x 1 6 ln x 2 2 x 2 arc tg x 1 c 5

37.-

2 x

2

5 x 5 dx 3 x 2 x 2 3x 2 632

x

SOLUCIN: 5 x 5 dx x 2 5 x 5 dx ! 2 x 2 3x 2 x 2 3x 2 63 2 x 3x 3 2 x 2 3x 7 x 2 5 x 5 dx A B Cx D ! 2 2 2 x 3x 3 2 x 3x 7 2 x 3 x 3 2 x 3 x 7 x 2 5 x 5 ! 2 A 4 B 2C x 3 9 A 3C 2 D x 2 A 5 B 3 D 9C x 16 21 A 21B 9C 2

x

97

UNAC-FIEE2 A 4 B 2C ! 0 9 A 3C 2 D ! 1 16 A 5 B 3 D 9C ! 5 21A 21B 9C ! 5 A ! 16 / 135 B ! 19 / 144 C ! 413 / 1080 D ! 1157 / 720

MATEMTICA II

2 x

!

3 7 413 19 8 ln x 2 x ln x 3 ln 2 x 3 2 2 4320 144 135 1901 4x 3 47 .arc tg 720 47

B) CALCULAR LAS SIGUIENTES INTEGRALES RACIONALES DE SENO Y 1.-

3 5 cos xSOLUCIN:tg x !t 2 dx !

2dt x 2 tg 2 dx dt 1 2t 1 1 t 2 c 3 5 cos x ! 5 1 t 2 ! 4 t 2 ! 4 Ln 2 t c ! 4 Ln x 2 tg 3 2 1 t 2

2.-

1 cos x dxSOLUCIN:2dt 2 dx cos x 1 ! x 1 t 2 ! dx ! 1 dx ! x 1 cos x 1 t 1 cos x 1 cos x 1 1 t2 x ! x dt ! x t c ! x tg c 2

cos x

3.-

senx cos x98

)2

5 x 5 dx 16 dx 19 dx ! 2 2 x 3 144 x 3 3x 2 x 3 x 2 63 1352

x

826 x 3471dx 1 2 x 2 3x 7 2160

COSENO.

dx

1 t 2 2 dt , cos ! 1 t 2 1 t 2

dx

UNAC-FIEE SOLUCIN:2dt dx dt 1 t 2 senx cos x ! 2t 1 t 2 ! 2 2t 1 t 2 ! 1 t 2 1 t 2 ! 2 2 2 Ln

MATEMTICA II

dt

2 t 12

2

!

2 t 1 1 c! Ln 2 t 1 2

x 1 2 c x 2 tg 1 2 2 tg

4.-

1 senx dxSOLUCIN: senx sen 2 x senx senx senx 1 dx ! dx ! dx ! 1 senx 1 sen 2 x cos 2 x ! tgx sec x tg 2 x dx ! sec x tgx x c

senx

5.-

8 4 senx 7 cos xSOLUCIN:2dt 1 t2 8 7t 8t 7 2 1 t 2 1 t2

dx

8 4 senx 7 cos x !

dx

! 2

dt ! t 8t 52

x 5 dt t 4 1 2 ! ! Ln c ! Ln c x t 4 1 t 42 1 tg 3 2 tg

6.-

cos x 2senx 3SOLUCIN:2 dt 1 t 2 1 t 4t 3 2 1 t 1 t 22

dx

cos x 2senx 3 !

dx

! 2

dt dt ! 2 t 2t 2 2t 4t 42

99

UNAC-FIEE ! 7.-

MATEMTICA II dt x arc tg t 1 c ! arc tg tg 1 c 2 t 1 1 2

1 senx cos x dxSOLUCIN:

1 senx cos x

1 senx cos x dx ! 1 1 senx cos x dx ! x 2 1 senx cos x COMO: senx !2t 1 t 2

1 senx cos x

2

dx

cos x !

1 t2 1 t 2

dx !

2 dt 1 t 2

2dt dt 1 1 1 t2 ! ! dt ! 2 2 2t 1 t t t t t 1 1 1 t2 1 t2 x tg t 2 ! Ln ! Ln x t 1 tg 1 2 x tg 1 senx cos x 2 1 senx cos x dx ! x 2 Ln x c tg 1 2 dx 1 senx cos x !

8.-

2senx 3 cos x 5SOLUCIN:2 dt dx dt 1 t 2 ! ! 2 2senx 3 cos x 5 2 4t 2t 1 2t 1 t 3 2 2 1 t 2 5 1 t 1 t 1 1 dt 4 ! 1 arc tg 4t 1 c ! ! arc tg 2 3 4 1 2 3 3 3 3 t 4 4 4 x 4tg 1 1 2 c arc tg 3 3

dx

100

UNAC-FIEE

MATEMTICA II

9.-

1 tgx dxSOLUCIN:dt 1 t 2 1 tgx 1 t dt dt tdt 1 tgx dx ! 1 t 1 t 2 ! 1 t 1 t 2 ! 1 1 ! Ln t 1 Ln t 2 1 c ! Ln sec 2 x Ln tgx 1 c 2 2

1 tgx

SEA tgx ! t

dx !

10.-

1 cos x dx3

senx

SOLUCIN: Q ! 1 cos x dQ ! senx dx SEA senxdx dQ 1 1 1 cos x 3 ! Q 3 ! 2Q 2 c ! 21 cos x c 11.-

1 sen 1 sen

sen 2 x2

x

dx

SOLUCIN:2 senx. cos xdx 1 sen 2 x x SEA Q ! 1 sen 2 x dQ ! 2 senx. cos xdx dQ sen2 x 2 1 sen 2 x dx ! Q ! Ln Q c ! Ln 1 sen x c2

sen 2 x

dx !

12.-

2 senx3 senx SOLUCIN: SEA t ! senx 1 !

dx

A B 2 senx 3 senx 2 t 3 t 1 ! A3 t B2 t 1 ! A B t 3 A 2 B A B ! 0 A ! 1 , B ! 1 3 A 2 B ! 1

101

UNAC-FIEE

MATEMTICA II

2 dt 2 dt 2 2 1 1 dx 1 t ! 1 t dx ! 2 senx 3 senx 2 senx 3 senx 2dt 2t 2 3 2 1 t 1 t 2 x x tg 1 1 3tg 1 2 dt 2 dt 2 c ! 2 ! arc tg 2 arc tg t t 3t 2 2t 3 3 3 2 2 2

13.-

2 cos x 3 cos x SOLUCIN: SEA z ! cos x 1 !

dx

A B 2 cos x 3 cos x 2 z 3 z 1 ! A3 z B2 z 1 ! A B z 3 A 2 B A B ! 0 A ! 1 , B ! 1 3 A 2 B ! 0 2 dt 2 dt 2 1 1 1 1 t 2 1 t 2 cos x 3 cos x ! 2 cos x 3 cos x dx ! 1 t 2 1 t 2 3 2 1 t 2 1 t 2 x x tg 1 tg dt dt 2 2 arc tg arc tg 2 c ! 2 2 ! t 3 t2 2 3 2 3 2 14.-

x x 2

x2

2

1 dx

1 x 4 1

SOLUCIN: B x dx x 1dx. x A x 2 x 2 1 x 4 1 ! x 2 x 2 1 x 4 1 x2 1 B A ! 2 2 2 2 x x 1 x x 1

x 2 1 ! Ax 2 A Bx 2 p A ! 1 x 2 1 ! A B x 2 A p B ! 2 x dx x dx 2 x dx 1 x 2 x 2 1 x 4 1 ! x 2 x 4 1 2 x 2 1 x 4 1 HACIENDO

102

UNAC-FIEE

MATEMTICA II

tan E ! x 2 sec 2 dx ! 2 xdx 1 cos E ! x4 1

E ! Iy !

sec 2 E dE . cos E sec 2 E dE . cos E 2 2tan E 1 2 tan E

1 dE csc E dE senE cos E 2 1 dE I y ! ln cscE ctgE 2 2senT / 4 cosT / 4 E pero ! senE cos E ! 2 senT / 4cosT / 4 E 1 1 I y ! ln csc E ctgE secT / 4 E dE 2 2 senT / 4

1 ln secT / 4 E tag T / 4 E 2senT / 4 1 1 sen T / 4 E ln 2 sen T / 4 cos T / 4 E

1 senT / 4 cos E senE cosT / 4 1 1 I y ! ln csc E ctgE ln cos T / 4 cos E senT / 4.senE 2 senT / 4 2 1 senT / 4cos E senE cosT / 4 1 1 I y ! ln csc E ctgE ln cos T / 4 cos E senT / 4.senE 2senT / 4 21 I y ! ln 2 1 2 2 1 x2 x4 1 ln x2 2 2 1 x2 1

15.-

x arc sen x dx3

SOLUCIN: 1 dx arc sen x ! Q dQ ! x x 2 1 4 3 dx ! dv v ! x x 4 4 x x4 dx 1 1 x 3 arc sen dx ! arc sen x x 4 4 x x2 1

103

UNAC-FIEEx 3 dx x2 1 sec 3 U . secU .tgU .dU ! sec 4 U dU tgU

MATEMTICA II

!

! 1 tg 2U sec 2 U dU ! sec 2 U dU tg 2U . sec 2 U dU ! tgU 3

tg 3U x 2 1 c ! x 2 1 c 3 3x 2 1 c 33

1 1 x4 2 x arc sen x dx ! 4 arc sen x x 1

16.-

a cos x bsenxSOLUCIN:2 dt 2dt 2 1 t2 ! ! 2 1 t2 2t a at 2 dt a a 2 b 2 a b 2 2 1 t 1 t a a2 b2 b t 2 1 a a ! Ln c a a2 b2 a2 b2 b t 2 a a a x a 2 b 2 atg b 1 2 ! Ln c 2 2 x 2 2 a b a b atg b 2

dx

dt b 2 t a 2

17.-

a

2

dx sen x b 2 cos 2 x2

SOLUCIN: dx sec 2 xdx ! 2 2 a 2 sen 2 x b 2 cos 2 x a tg x b 2 SEA tgx ! Q sec 2 xdx ! dQ 1 1 1 adQ aQ a ! ! arc tg arc tg tgx c c ! 2 2 a aQ b ab ab b b 18.-

tgx 1sen

dx

2

x SOLUCIN:

104

UNAC-FIEE z ! tgx dx ! dz 1 z2 ; senz ! z 1 z2

MATEMTICA II

dz dz Bz C A 1 z2 ! ! dz 2 2 1 z z z z2 1 z 1 z 2 1 z A B z 2 B C z C ! 1 A A B!0 !1 B C ! 0 B ! 1 !1 !1 C C

1 z z

dz

2

z 1dz ! ln 1 z ln z 1 dz z 1 1 ! dz ! 2 z2 z z 1 z 1 z 1 tgx 1 ! ln c tgx tgxSOLUCIN:

19.-

4 tgx 4ctgx

dx

C) CALCULAR LAS SIGUIENTES INTEGRALES IRRACIONALES 1.-

x 3 dx x 1 SOLUCIN:

SEA COMOx3

z 2 ! x 1 dx ! 2 z dz z2 ! x 1 dx ! z! x 1

z

2

x 1

1 2 z dz ! z3

z 6 3z 4 3 z 2 1 2 z dz z

z7 3 z ! 2 6 3 z 4 3 z 2 1dz ! 2 z 5 z 3 z c 7 5 3 6 x 1 3 z 3 2 x 1 x c ! 2 z z 4 z 2 1 c ! 2 x 1 7 5 7 5

2.-

3

x dx a