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Evaluate of each of the following integr...

Evaluate of each of the following integral: `int_(-pi//2)^(pi//2)(cos^2x)/(1+e^x)dx`

Text Solution

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Let I=`int_(-pi//2)^(pi//2)(cosx)/(1+e^x)dx...(1)`
I=`int_(-pi//2)^(pi//2)(cos(pi/2-pi/2-x))/(1+e^(pi/2-pi/2-x))dx`
I=`int_(-pi//2)^(pi//2)(cos(x))/(1+e^(-x))dx`
I=`int_(-pi//2)^(pi//2)(e^xcos(x))/(1+e^(x))dx...(2)`
(1)+(2)
`2I=int_(-pi//2)^(pi//2) cosxdx`
`2I=2int_(0)^(pi//2) cosxdx`
`2I=2`
...
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