For any integer n, the integral _0^ e^ ^2 x ^3 (2n + 1)x ,dx = - Vidyadip

MCQ

For any integer $n,$ the integral $\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $

A
$ - 1$
✓
$0$
C
$1$
D
$\pi $

✓

Answer

Correct option: B.

$0$

b
(b) Let $f(x) = \int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x.dx} $

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

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

Hence by the property of definite integral,

$\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = 0} $, $[f(2a - x) = - f(x)]$.

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