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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`

Text Solution

Verified by Experts

The correct Answer is:
`pi//2`
Let `I=int_(-pi)^(pi)(cos^(2)x)/(1+a^(x))dx, agt0` ………….. 1
`:.I=int_(-pi)^(pi)(cos^(2)x)/(1+a^(-x))dx` (Replacing `x` by `-x`)
or `I=int_(-pi)^(pi)(a^(x)cos^(2)x)/(1+a^(x))dx`…………….2
Adding 1 and 2 we get
`2I=4int_(0)^(pi//2)cos^(2)x dx`
`=2int_(0)^(pi//2)(1+cos2x)dx`
`=2[x+(sin2x)/2]_(0)^(pi//2)`
`-pi`
or `I=(pi)/2`
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