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Evaluate int(0)^(pi)(e^(cosx))/(e^(cosx)...

Evaluate `int_(0)^(pi)(e^(cosx))/(e^(cosx)+e^(-cosx))dx`.

Text Solution

Verified by Experts

`I=int_(0)^(pi)(e^(cosx))/(e^(cosx)+e^(-cosx))dx` ……….`(i)`
`=int_(0)^(pi)(e^(cos(pi-x)))/(e^(cos(pi-x))+e^(-cos(pi-x)))dx`
`:.l=int_(0)^(pi)(e^(-cos(pi-x)))/(e^(-cosx)+e^(cosx))dx`……..`(ii)` `(:' cosx(pi-x)=-cosx)`
Adding `(i)` and `(ii)`,
`2l=int_(0)^(pi)(e^(cosx)+e^(-cosx))/(e^(cosx)+e^(-cosx))dx=int_(0)^(pi)dx=pi`
`:.l=(pi)/(2)`
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