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Question: The value of the integral \(\int _ { - \pi } ^ { \pi } \sin m x \sin n x d x\) for \(m \neq n\) ...

The value of the integral ∫−ππsin⁡mxsin⁡nxdx\int _ { - \pi } ^ { \pi } \sin m x \sin n x d x for m≠nm \neq n (m,n∈I)( m , n \in I ) is

A

0

B

π\pi

C

π2\frac { \pi } { 2 }

D

2π2 \pi

Answer

0

Explanation

Solution

Let I=2∫0πsin⁡mxsin⁡nxdx=∫0π[cos⁡(m−n)x−cos⁡(m+n)x]dxI = 2 \int _ { 0 } ^ { \pi } \sin m x \sin n x d x = \int _ { 0 } ^ { \pi } [ \cos ( m - n ) x - \cos ( m + n ) x ] d x

=[sin⁡(m−n)x(m−n)−sin⁡(m+n)x(m+n)]0π\left[ \frac { \sin ( m - n ) x } { ( m - n ) } - \frac { \sin ( m + n ) x } { ( m + n ) } \right] _ { 0 } ^ { \pi }

=[sin⁡(m−n)π(m−n)−sin⁡(m+n)π(m+n)]=0= \left[ \frac { \sin ( m - n ) \pi } { ( m - n ) } - \frac { \sin ( m + n ) \pi } { ( m + n ) } \right] = 0 .

Since, sin⁡(m−n)π=0=sin⁡(m+n)π\sin ( m - n ) \pi = 0 = \sin ( m + n ) \pi for m≠nm \neq n.