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inte^(x)((1+sinx)/(1+cosx))dx=?...

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

A

`e^(x)"sin"(x)/(2)+C`

B

`e^(x)"cos"(x)/(2)+C`

C

`e^(x)"tan"(x)/(2)+C`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`I=inte^(x)((1+2sin((x)/(2))cos((x)/(2)))/(2cos^(2)((x)/(2))))dx=inte^(x)("tan"(x)/(2)+(1)/(2)"sec"^(2)(x)/(2))dx`
`=inte^(x){f(x)+f'(x)}dx," where "f(x)="tan"(x)/(2)`
`=e^(x)f(x)+C=e^(x)"tan"(x)/(2)+C`.
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