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

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

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int_(0)^(pi) e^(cos^(2)x)cos^(3)(2n+1)x dx, n epsilonZ` …………..1
`=int_(0)^(pi) e^(cos^(2)(pi-x))cos^(3)[(2n+1)(pi-x)]dx`
[Using `int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx`]
`=int_(0)^(pi)e^(cos^(2)x)cos^(3)[(2n+1)pi-(2n+1)x]dx`
`=-int_(0)^(pi)-e^(cos^(2)x)cos^(3)(2n+1)xdx`
`=-I`
or `I=0`
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