First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{π} \sin^{n} x \cos^{2 m + 1} x d x$ is

Easy

A

$\frac{(2 m + 1)!}{(n!)^{2}}$
B

$\frac{(2 m + 1)!}{n!}$
C

$\int_{0}^{π} \cos^{2 m - 1} x d x$
D

none of these

Solution

$I = \int_{0}^{π} \sin^{n} x \cos^{2 m + 1} x d x$
$\begin{array}{l} = \int_{0}^{π} \sin^{n} (π - x) \cos^{2 m + 1} (π - x) d x \\ = - \int_{0}^{π} \sin^{n} x \cos^{2 m + 1} x d x = - I \end{array}$
So $2 I = 0$ which implies that $I = 0$ Also, $\int_{0}^{π} \cos^{2 m - 1} x d x = 0$
by a similar reasoning.

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