`int_(pi//2)^(pi//2)(cosx)/(1+e^(x))dx` is equal to

Options

A. 1
B. 0
C. -1
D. None of these

Answer

A

Explanation

`I-int _(-pi)^(pi//2)(cos x)/(1+e^(x))dx" (i)"`
`I=int _(-pi)^(pi//2)(cos (pi//2-pi //2-x))/(1+e^((x//2-pi//2-x)))dx`
`=int _(-pi)^(pi//2)(cos (-x))/(1+e^(-x))dx`
`=int _(-pi)^(pi//2)^(e^(x)cos x)/(1+e^(x))dx`
`=int _(-pi)^(pi//2)(e^(x)cos x)/(1+e^(x))dx" "(ii)`
On adding Eqs. (i) and (ii), we get
`2I=int_(-pi//2)^(pi//2)((1+e^(x))cos )/((1+e^(x)))dx`
`int _(-pi//2)^(pi//2)cos x dx`
`=2int _(0)^(pi//2)cos x dx`
[Since, cos x is an even function.]
`therefore 2I=2[sin x]_(0)^(pi//2)=2(1-0)=2`
`impliesI=1`