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The value of the integral int(-pi)^(pi)(...

The value of the integral `int_(-pi)^(pi)(cos^(2)x)/(1+a^(x))"dx"`, where `a gt 0`, is - (A) ` pi ` (B) ` api ` (C) ` pi/2 ` (D) ` 2pi `

A

`pi`

B

`api`

C

`pi//2`

D

`2pi`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int_(-pi)^(pi)(co^(2)x)/(1+a^(x))"dx…………(1) "int_(a)^(d)f(x)dx=int_(a)^(d)f(a+b-x)dx`
`I=int_(-pi)^(pi)(cos^(2)(-x))/(1+a^(-x))dx`
`I=int_(-pi)^(pi)(a^(x)cos^(2)x)/(1+a^(x))"dx………….(2)"`
add equation (1) and (2)
`2I=int_(-pi)^(pi)cos^(2)x((1)/(1+a^(2))+(a^(x))/(1+a^(x)))"dx"`
`I=int_(0)^(pi)cos^(2)"x dx = 2"int_(0)^(pi//2)cos^(2)"x dx"`
`=pi//2`
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