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

1.	For any integer `n`, the integral `int_0^pie^(cosx)cos^3(2n+1)xdx` has the value
Answer» `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` `=> I = -I` `=>2I = 0=> I = 0` `:. int_(0)^pi e^cosx cos^3(2n+1)x dx = 0`

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