For any integer $n,$ the integral $\int_{0}^{\pi}{e^{\cos^{2}x}\cos^{3}{(2n - 1)x\ dx}}$ has the value

Question

Dr. Abubacker · Accepted Answer

(c)
Given, $\int_{0}^{\pi}{e^{\cos^{2}x} \bullet \cos^{3}{(2n + 1)x\ dx}}$
Let $f(x) = e^{\cos^{2}x} \bullet \cos^{3}{(2n + 1)x\ }$
Then, $f(\pi - x) = e^{\cos^{2}{(\pi - x)}} \bullet {\cos^{3}\lbrack}{(2n + 1)\pi - (2n + 1)xbrack\ \ }$
$= - e^{\cos^{2}x} \bullet \cos^{3}{(2n + 1)x\ }$
$\Rightarrow \ f(\pi - x) = - f(x)$
Then, $f(x)$ is an odd function
$	herefore\ \int_{0}^{\pi}{e^{\cos^{2}x} \bullet \cos^{3}{(2n + 1)x\ dx}} = 0$

Q. For any integer $n,$ the integral $\int_{0}^{\pi}{e^{\cos^{2}x}\cos^{3}{(2n - 1)x\ dx}}$ has the value

Explanation

Reviewed by Dr. Abubacker

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