Guia de Ejercicios,

MM-202

Carlos Cruz

September 10, 2015

Nombre: Registro Estudiantil:

Instrucciones: Resuelva cada ejercicios de forma clara honesta y ordenada mostrando todo su proced-imiento de lo contrario no tendra opcion a creditos1

INTEGRACION POR PARTES

1.

sin(ln x)dx

(a) 12x cos(ln x) sin(ln x) + C (b) 1

2x cos(ln x) 1

2x sin(ln x) + C

(c) 12x cos(ln x) +

1

2x sin(lnx) + C (d) 1

2cos(ln x) +

1

2x sin(ln x) + C

(e) 12cos(ln x) +

1

2sin(lnx) + C

2.

cos(ln x)dx

(a) 12x cos(ln x) sin(ln x) + C (b) 1

2x cos(ln x) 1

2x sin(ln x) + C

(c) 12x cos(ln x) +

1

2x sin(lnx) + C (d) 1

2cos(ln x) +

1

2x sin(ln x) + C

(e) 12cos(ln x) +

1

2sin(lnx) + C

3.

x21 x2dx =

(a)1

2x1 x2 + 1

2sin1 x+ C (b) 1

2x1 x2 1

2sin1 x+ C

(c) 2x1 x2 + 2 sin1 x+ C (d) 1

2x1 x2 + 1

2sin1 x+ C

(e) x1 x2 + 1

2sin1 x+ C

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4.

a2 + x2dx, a R

(a)1

2xa2 + x2 +

a2

4ln(

a2 + x2 + x)+ C (b)

1

2xa2 + x2 + a2 ln

(a2 + x2 + x

)+ C

(c) xa2 + x2 + a2 ln

(a2 + x2 + x

)+ C (d) x

a2 + x2 +

a2

2ln(

a2 + x2 + x)+ C

(e) 2xa2 + x2 +

a2

2ln(

a2 + x2 + x)+ C

5.

x2ex

(x+ 2)2dx

(a)ex(x 2)x+ 2

+ C (b)ex(x+ 2)

x 2 + C (c)ex(x 2)(x+ 2)2

+ C (d)2ex(x 2)x+ 2

+ C

(e)(x 2)ex(x+ 2)

+ C

6.

(sin1 x

)2dx

(a) 21 x2 sin1 x 2x+ x2 sin1 x+ C (b) 2

1 x2 sin1 x 1

2x+ x(sin1 x)2 + C

(c)1 x2 sin1 x 2x+ x(sin1 x)2 + C (d) 2

1 x2 sin1 x 2x+ x(sin1 x)2 + C

(e) 21 x2 sin x 2x+ x(sin1 x)2 + C

7.

xn ln xdx, n 6= 1

(a)xn+1 ln x

n+ 1 x

n+2

(n + 1)(n+ 2)+ C (b)

xn+1 ln x

n + 1 x

n+2

(n+ 1)2+ C

(c)xn+1 ln x

n + 1+

xn+1

(n + 1)2+ C (d)

xn+1 ln x

n + 1 x

n+1

(n+ 1)2+ C

(e)xn ln x

n + 1 x

n+2

(n+ 1)(n+ 2)+ C

8.

tan1(

x)dx

(a) (x+ 1) tan(x)x+ C (b) (x+ 1) tan1(x)x+ C

(c) (x 1) tan1(x)x+ C (d) (x+ 1) tan1(x) +x+ C

(e) x tan1(x)x+ C

9.

sin1 x1 xdx

(a)1 x sin1 x+

1 + x+ C (b)

1 x sin1 x+ 4

1 + x+ C

(c) 21 x sin1 x+1 + x+ C (d) 21 x sin1 x+ 41 + x+ C

(e) 21 x sin1 x+ 41 + x+ C

2

10.

x tan1 x

1 + x2dx

(a)1 + x tan1 x ln(x+

1 + x2) + C (b)

1 + x2 tan1 x+ ln(x+

1 + x2) + C

(c)1 + x2 tan1 x ln(x+

1 + x2) + C (d)

1 + x2 tan1 x x ln(x+

1 + x2) + C

(e)1 + x2 tan1 x 1

2ln(x+

1 + x2) + C

11.

sin1(

x)

1 x dx

(a) 2x 21 x sin1(x) + C (b) 2x1 x sin1(x) + C

(c) 2x 1

2

1 x sin1(x) + C (d) 2x 2

1 x sin(x) + C

(e)x 21 x sin1(x) + C

12.

exdx

(a) 2ex(x 1) + C (b) 1

2ex(x 1) + C

(c) ex(x 1) + C (d) 2e

x(x 1) + C

(e) 2ex(x 2) + C

13.

sin( 3

x)dx

(a) 3 3x sin( 3

x) 3( 3

x2 2) cos( 3x) + C (b) 6 3x sin( 3x) 3( 3

x2 2) cos( 3x) + C

(c) 6 3x sin( 3

x) ( 3

x2 2) cos( 3x) + C (d) 6 3x sin(x) 3( 3

x2 2) cos(x) + C

(e) 6 3x sin( 3

x) 3( 3

x2 3) cos( 3x) + C

14.

sin1 x(1 x2)3dx

(a)x sin1 x1 x2 + ln(1 x

2) + C (b)sin1 x1 x2 + ln(1 x

2) + C

(c)x sin1 x1 x2 ln(1 x

2) + C (d) 2x sin1 x1 x2 + ln(1 x

2) + C

(e)x sin1 x1 x2 + 2 ln(1 x

2) + C

15.

x2 tan1 x

1 + x2dx

(a) 12ln(x2 + 1) 1

2(tan1 x)2 + x tan1 x+ C (b)

1

2ln(x2 + 1) 1

2(tan1 x)2 + x tan1 x+ C

(c) 14ln(x2 + 1) 1

2(tan1 x)2 + x tan1 x+ C (d) 1

2ln(x2 + 1) 1

2(tan1 x)2 + x tan1 2x+ C

(e) ln(x2 + 1) 12(tan1 x)2 + x tan1 x+ C

3

16.

tan1 x

x2(1 + x2)dx

(a) 12ln(1 + x2) + ln x 1

2(tan1 x)2 tan

1 x

x+ C

(b) 12ln(1 + x2) + lnx 1

2(tan1 x)2 tan1 x+ C

(c)1

2ln(1 + x2) + ln x 1

2(tan1 x)2 tan

1 x

x+ C

(d) 12ln(1 + x2) + lnx+

1

2(tan1 x)2 tan

1 x

x+ C

(e)1

2ln(1 + x2) ln x+ 1

2(tan1 x)2 +

tan1 x

x+ C

17.

(x2 x+ 1)ex

(x2 + 1)3

2

dx

(a)ex

(x2 + 1)3

2

+ C (b)ex

(x2 + 1)2+ C (c)

ex

x2 + 1+ C (d)

exx2 + 1

+ C (e)xexx2 + 1

+ C

18.

x3ex

2

(x2 + 1)2dx

(a) ex2

x2 + 1+ C (b) 1

2

ex2

x2 + 1+ C (c)

ex2

x2 + 1+ C (d)

1

2

ex2

x2 + 1+ C (e)

1

2

ex

x2 + 1+ C

19.

ex(x 1)

x2dx

(a)ex

x(b) e

x

x+ C (c)

ex

x+ C (d)

ex

x2+ C (e)

ex

x+ C

20.

ln(x+

x2 a2

)dx, a R

(a) x ln(x+

x2 a2

)x2 + a2 + C (b) x ln

(x+

x2 a2

)+x2 a2 + C

(c) x+ ln(x+

x2 a2

)x2 a2 + C (d) x ln

(x+

x2 a2

)x2 a2 + C

(e) x ln(x+

x2 a2

)x a+ C

EJERCICIOS DIVERSOS: Utilizando integracion por partes resuelva las siguientes integrales:

1.

sin1(

x)

xdx 2.

x sin1(3x2)dx 3.

x

+ xdx, , R

4.

xex

(1 + x)2dx 5.

cos1

(x

x+ 1

)dx

6.

ln(x+

a2 + x2

)dx 7.

ln x 1(ln x)2

dx 8.

x ln x(x2 1)3dx

9.

xex1 + ex

dx 10.

x tan1 x

(1 + x2)2dx

4

FORMULAS DE REDUCCION USANDO INTEGRACION POR PARTES: Usando inte-gracion por partes demuestre la validez de las siguientes integrales

1.

xn cos xdx = xn sin x n

xn1 sin x

2.

1

(a2 + x2)mdx =

1

a2

(x

(2m 2)(a2 + x2)m1 +2m 32m 2

1

(a2 + x2)m1dx

), m 6= 1

3.

sinm xdx = sin

m1 x cosx

m+m 1m

sinm2 xdx

4.

sinm x cosn xdx =

sinm+1 x cosn1 x

m+ n+

n 1m+ n

sinm x cosn2 xdx, m 6= n

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