If n is any integer, then _0^ e^ ^2 x ^3 (2n + 1)x ,dx = - Vidyadip

Question

If $n$ is any integer, then $\int_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $

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Answer

c
(c) Since $\cos (2n + 1)(\pi - x) = \cos [(2n + 1)\pi $$ - (2n + 1)x]$

$= - \cos (2n + 1)x$ and ${\cos ^2}(\pi - x) = {\cos ^2}x$

So that $f(2a - x) = - f(x)$, and

hence by the property of definite integral

$\int_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}(2n + 1)x\,dx = 0} $.

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