Question

For any integer `n`, the integral `int_0^pie^(cos^x)cos^3(2n+1)xdx` has the value
A. `pi`
B. 1
C. 0
D. None of these

Answer 1

Correct Answer - C
Let `I=int_(0)^(pi)e^(cos^(2)x)*cos^(3){(2n+1)xdx`
Using `int_(0)^(a)f(x)dx={{:(" "0_(,),f(a-x)=-f(x)),(2int_(0)^(a)f(x)dx_(,),f(a-x)=f(x)):}` Again , let f(x) `=e^(cos^(2)x)*cos^(3){(2n+1)x}`
`:.f(pi-x)=(e^(cos^(2)x){-cos^(3)(2n+1)x}=-f(x)`
`:.I=0`