$\int\limits_{-\pi}^\pi \sin m x \sin n

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Definite Integration

Question:

$\int\limits_{-\pi}^\pi \sin m x \sin n x d x (m \neq n$ and m, n are integers) =

Options:

$\pi$

$\pi / 2$

$2 \pi$

Correct Answer:

Explanation:

$I=\frac{1}{2} \int\limits_0^\pi 2 \sin m x \cos n x d x$

$=\frac{1}{2} \int\limits_0^\pi \cos (m-n) x-\cos (m+n) x d x$

$=\frac{1}{2}\left[\frac{\sin (m-n) x}{m-n}-\frac{\sin (m+n) x}{m+n}\right]_0^\pi=0$