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Evaluate int(-pi)^(pi)(cos^(2)x)/(1+a^(x...

Evaluate `int_(-pi)^(pi)(cos^(2)x)/(1+a^(x))dx`

A

`pi`

B

`api`

C

`(pi)/(2)`

D

`2pi`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `I=int_(-pi)^(pi)(cos^(2)x)/(1+a^(x))dx` . . . (i)
`=int_(pi)^(-pi)(cos^(2)(-x))/(1+a^(-x))d(-x)`
`rArrI=int_(-pi)^(pi)a^(x)(cos^(2)x)/(1+a)dx`
On adding Eqs . (i) and (ii) , we get
`2I=int_(-pi)^(pi)((1+a^(x))/(1+a^(x)))cos^(2)xdx`
`=int_(-pi)^(pi)cos^(2)xdx=2int_(0)^(pi)(1+cos2x)/(2)dx`
`=int_(0)^(pi)(1+cos2x)dx`
`=int_(0)^(pi)1dx+int_(0)^(pi)cos2xdx`
`=[x]_(0)^(pi)+2int_(0)^(pi//2)cos2xdx=pi+0`
`rArr2I=pirArrI=pi//2`
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