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

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

A

`pi`

B

1

C

0

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
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`
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