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int(pi//2)^(pi//2)(cosx)/(1+e^(x))dx is ...

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

A

1

B

0

C

`-1`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`l = int_(-pi//2)^(pi//2)(cos x)/(1+e^(x))dx` ….(i)
`rArr l = int_(-pi//2)^(pi//2)(cos((pi)/(2)-(pi)/(2))-x)/(1+e^((pi//2-pi//2-x)))dx`
`= int_(-pi//2)^(pi//2)(cos(-x))/(1+e^(-x))dx`
`= int_(-pi//2)^(pi//2)(cos x)/(1+e^(-x))dx = int_(-pi//2)^(pi//2)(e^(x) cos x)/(1+e^(x))dx` ....(ii)
On adding Eqs. (i) and (ii), we get
`2l = int_(-pi//2)^(pi//2)((1+e^(x))cos x)/((1+e^(x)))dx = int_(-pi//2)^(pi//2)cos xdx`
` = 2 int_(0)^(pi//2)cos xdx " " [because " cos x is an even function"]`
`rArr 2l = 2[sin x]_(0)^(pi//2)=2(1-0)=2 rArr l = 1`
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