Q. int0^pie^(cos^2x)( cos^3(2n+1) x dx, ...

Q. `int_0^pie^(cos^2x)( cos^3(2n+1) x dx, n in I`

`pi`

`1`

`0`

none of these

Text Solution

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The correct Answer is:

`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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