`inte^x((1+sinx)/(1+cosx))dx`

Options

A. ` ( 1 ) / ( 2 ) e ^ x sec "" ( x )/ ( 2 ) + C `
B. ` e ^ x sec "" ( x ) / ( 2 ) + C `
C. ` (1 ) / ( 2 ) e ^ x tan "" ( x ) / ( 2 ) + C `
D. ` e ^ x tan "" ( x ) / ( 2 ) + C `

Answer

D

Explanation

Let I ` = int e ^ x [ ( 1 + sin x ) /( 1 + cos x ) ] dx `
` = int { ( e^ x )/( ( 1 + cos x ) ) + ( e ^ x sin x ) /( ( 1 + cos x )) } dx `
=` int ( e^ x ) /( 2 cos ^ 2 "" ( x ) /( 2 ) ) dx + int ( e ^ x * 2 sin "" ( x )/ ( 2 ) * cos "" ( x )/ ( 2 ) ) /( 2 cos ^ 2 "" ( x ) / ( 2 ) ) * dx `
` = ( 1 )/ ( 2 ) int e^ x * sec^ 2 "" ( x ) /(2 ) * dx + int underset (II)(e^ x ) underset(I) ( tan "" ( x ) /( 2)) * dx `
` = ( 1 )/( 2 ) int e ^ x * sec ^ 2 "" ( x ) / ( 2 ) * dx +{ tan "" ( x ) /(2 )* e^ x- int ( 1 ) / ( 2 ) * sec^ 2 "" ( x )/ ( 2 ) * e ^ x dx} `
( using integral by parts )
` = ( 1 )/( 2 ) int e ^ x * sec^ 2 "" ( x ) /( 2 ) dx + e ^ x * tan "" ( x ) /( 2 ) - ( 1 ) / (2 ) `
` int e ^ x * sec ^ 2 "" ( x ) /( 2 ) * dx `
` = e^ x * tan "" (x )/ ( 2 ) + C `