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

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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