Definite Integration Question 387
Question: The value of the integral $ \int _{-\pi }^{\pi }{\sin mx\sin nxdx} $ for $ m\ne n $
$ (m,n\in I), $ is
Options:
A) 0
B) $ \pi $
C) $ \frac{\pi }{2} $
D) $ 2\pi $
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ I=2\int_0^{\pi }{\sin mx\sin nxdx}=\int_0^{\pi }{[\cos (m-n)x-\cos (m+n)x]dx} $
= $ [ \frac{\sin (m-n)x}{(m-n)}-\frac{\sin (m+n)x}{(m+n)} ]_0^{\pi } $
$ =[ \frac{\sin (m-n)\pi }{(m-n)}-\frac{\sin (m+n)\pi }{(m+n)} ]=0 $ . Since, $ \sin (m-n)\pi =0=\sin (m+n)\pi $ for $ m\ne n $ .
BETA
