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For any integer n, the integral int0^pie...

For any integer `n`, the integral `int_0^pie^(cosx)cos^3(2n+1)xdx` has the value

Text Solution

Verified by Experts

`I = int_(0)^pi e^cosx cos^3(2n+1)x dx`
We know,
` int_(0)^a f(x)dx = int_(0)^a f(a-x)dx `
`:. I = int_(0)^pi e^cos(pi-x )cos^3(2n+1)(pi-x) dx`
`=> I = int_(0)^pi e^cos x cos^3[(2n+1)pi - (2n+1)x] dx`
`=> I = int_(0)^pi e^cos x cos^3[2npi+pi -(2n+1)x]dx`
`=>I = int_(0)^pi e^cos x cos^3[pi -(2n+1)x]dx`
`=> I = - int_(0)^pi e^cosx cos^3(2n+1)x dx`
...
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