Q. For any integer $ n $ , the integral $ \int\limits_{0}^{\pi}e^{\cos^2x}\cos^{3}\left(2n+1\right)x\, dx $ has the value

AMUAMU 2015Integrals Report Error

$ 1 $

$ \pi $

$ 2\pi $

None of these

Solution:

Let $f(x)=e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x$
Then, $f(\pi-x)=e^{\cos ^{2}(\pi-x)} \cdot \cos ^{3}[(2 n+1) \pi-(2 n+1) x]$
$=-e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x$
$\Rightarrow f(\pi-x)=-f(x)$
Hence, $\int\limits_{0}^{\pi} e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x d x=0$
$\left[\because \int\limits_{0}^{2 a} f(x) d x=0\right.$, if $\left.f(2 a-x)=-f(x)\right]$

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