UNIVERSIDADAUTONOMACHAPINGOPREPARATORIAAGRICOLAAREADEMATEMATICASCALCULOINTEGRAL Examenparcial1 10demarzode2012SOLUCIONValordecadaproblema: puntos1. Resolverlasiguienteintegraltrigonometrica_1cos (x) + 1dxSoluci on:1cos (x) + 1=1cos (x) + 1cos (x) 1cos (x) 1=1(cos (x) 1)(cos (x) + 1)(cos (x) 1)=cos (x) 1cos (x)21utilizandolaidentidad,sin (x)2+ cos (x)2= 1sin (x)2= cos (x)21setiene,1cos (x) + 1=cos (x) 1cos (x)21= cos (x) 1sin (x)2=1sin (x)2 cos (x)sin (x)2laprimeraintegralesinmediadaylasegundaseresuelveporcambiodevariable_1cos (x) + 1dx =_1sin (x)2 cos (x)sin (x)2dx=_1sin (x)2dx _cos (x)sin (x)2dx=_csc (x)2dx _cos (x)sin (x)2dx= csc (x) cot (x)1 geTEX2. Resolverlasiguienteintegraltrigonometrica_sec (3 y)3tan (3 y)3dysec (3 y)3tan (3 y)3= sec (3 y)2tan (3 y)2sec (3 y)tan (3 y)= sec (3 y)2 _sec (3 y)21_sec (3 y)tan (3 y)= sec (3 y)4sec (3 y)tan (3 y) sec (3 y)2sec (3 y)tan (3 y)laintegralsepuedeexpresarcomo,_sec (3 y)3tan (3 y)3dy==_sec (3 y)4sec (3 y)tan (3 y) sec (3 y)2sec (3 y)tan (3 y) dy=_sec (3 y)4sec (3 y)tan (3 y) dy +_sec (3 y)2sec (3 y)tan (3 y) dy=sec (3 y)515sec (3 y)393. Resolverlasiguienteintegraltrigonometrica_cos (x)5dxcos (x)5= cos (x)4cos (x)=_cos (x)2_2cos (x)= cos (x)_1 sin (x)2_2= cos (x)sin (x)42 cos (x)sin (x)2+ cos (x)_cos (x)5dx =_cos (x)sin (x)42 cos (x)sin (x)2+ cos (x) dx=_cos (x)sin (x)4dx 2_cos (x)sin (x)2dx +_cos (x) dx=sin (x)552 sin (x)33+ sin (x) + c4. Resolverlasiguienteintegraltrigonometrica_tan (2 x)4dxtan (2 x)4= tan (2 x)2tan (2 x)2=_sec (2 x)21_tan (2 x)22= sec (2 x)2tan (2 x)2tan (2 x)2laintegralseexpresacomo,_tan (2 x)4dx =_sec (2 x)2tan (2 x)2tan (2 x)2dx=_sec (2 x)2tan (2 x)2dx _tan (2 x)2dxu = tan (2 x)del (u) = 2 sec (2 x)2del (x)=_u2du2_sec (2 x)21dx= _sec (2 x)2dx + x +_u2du2= tan (2 x)2+ x +u36+ c=tan (2 x)36tan (2 x)2+ x + c5. Resolverlasiguienteintegraltrigonometrica_csc_x4_4dxSoluci on:csc_x4_4= csc_x4_2csc_x4_2=_1 + cot_x4_2_csc_x4_26. Resolverlasiguienteintegraltrigonometrica_sin (x)5_cos (x)dxSoluci on:sin (x)5_cos (x)= cos (x)12sin (x)5= cos (x)12sin (x)4sin (x)= cos (x)12_sin (x)2_2sin (x)3= cos (x)12_1 cos (x)2_2sin (x)= cos (x)92sin (x) 2 cos (x)52sin (x) +_cos (x) sin (x)_sin (x)5_cos (x)dx =_cos (x)92sin (x) 2 cos (x)52sin (x) +_cos (x) sin (x) dx=_cos (x)92sin (x) dx 2_cos (x)52sin (x) dx +_ _cos (x) sin (x) dx= 2 cos (x)11211+4 cos (x)7272 cos (x)3237. Resolverlasiguienteintegraltrigonometrica_cot (2 x)3csc (2 x) dxSoluci on:cot (2 x)3csc (2 x) = cot (2 x)2cot (x)csc (2 x)=_csc (2 x)21_cot (2 x)csc (2 x)= cot (2 x)csc (2 x)3cot (2 x)csc (2 x)_cot (2 x)3csc (2 x) dx =_cot (2 x)csc (2 x)3cot (2 x)csc (2 x) dx=_cot (2 x)csc (2 x)3dx _cot (2 x)csc (2 x) dx=csc (2 x)2csc (2 x)368. Resolverlasiguienteintegraltrigonometrica_sec (x)4tan (x)2dxSoluci on:sec (x)4tan (x)2= sec (x)2sec (x)2tan (x)2=_tan (x)2+ 1_sec (x)2tan (x)2= sec (x)2tan (x)4+ sec (x)2tan (x)2_sec (x)4tan (x)2dx =_sec (x)2tan (x)4+ sec (x)2tan (x)2dx=_sec (x)2tan (x)4dx +_sec (x)2tan (x)2dx=tan (x)55+tan (x)3349. Resolverlasiguienteintegraltrigonometrica_tan (y)3dySoluci on:tan (y)3= tan (y)2tan (y)=_sec (y)21_tan (y)= sec (y)2tan (y) tan (y)_tan (y)3dy=_sec (y)2tan (y) tan (y) dy=_sec (y)2tan (y) dy _tan (y) dy=tan (y)22log (sec (y))10. Resolverlasiguienteintegraltrigonometrica_cos (x)2sin (x)4dxSoluci on:cos (x)2sin (x)4=sin (x)4(cos (2 x) + 1)2=sin (x)4cos (2 x)2+sin (x)42=cos (6 x) 4 cos (4 x) + 7 cos (2 x) 432+cos (4 x) 4 cos (2 x) + 316=cos (6 x)32cos (4 x)16cos (2 x)32+116_cos (x)2sin (x)4dx =_cos (6 x)32cos (4 x)16cos (2 x)32+116dx=sin (6 x)192sin (4 x)64sin (2 x)64+x165