`int_(-pi/2)^(pi/2)(e^(|sinx|)cosx)/((1+e^(tanx))dxi se q u a lto` `e+1` (b) `1-e` `e-1` (d) none of these

Options

A. `e+1`
B. `2e`
C. `e-1`
D. `e-2`

Answer

C

Explanation

`I=int_(-pi//2)^(pi//2) (e^(|sinx|)/cosx)/(1+e^(tanx))dx` ……….1
`=int_(-pi//2)^(pi//2) (e^(|sin(-x)|)cos(-x))/(1+e^(tan(-x)))dx`
`:. I=int_(-pi//2)^(pi//2) (e^(tanx)e^(|sinx|)cosx)/(e^(tanx)+1)dx`………………2
Adding 1 and 2 we get
`2I=int_(-pi//2)^(pi/2) e^(|sinx|)cosx dx`
`=2int_(0)^(pi//2)e^(sinx)cosx dx`
`=2[e^(sinx)]_(0)^(pi//2) =2(e-1)`
`:. I=e-1`