`int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))dx,x in(0,(pi)/(2)) ` is equal to

Options

A. `log_(e)|1+e^(x)sinx|+c`
B. `log_(e)|e^(x)+cosx|+c`
C. `log_(e)|1+e^(x)cosx|-x+c`
D. `log_(e)|1+e^(x)cosx|+c`

Answer

D

Explanation

`int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))dx`
`=int(cos^(2)x-sin^(2)x)/((e^(-x)+cosx)sqrt((cosx+sinx)^(2)))dx`
`=int(cosx-sinx)/((e^(-x)+cosx))dx`
`=int(e^(x)(cosx-sinx))/(1+e^(x)cosx)dx`
`=int((1+e^(x)cosx)')/(1+e^(x)cosx)dx`
`=log_(e)|1+e^(x)cosx|+c`