Question
The value of $$\int_0^\pi {\frac{{\sin \,nx}}{{\sin \,x}}} dx,\,n\, \in \,N,$$ is :
A.
$$\pi $$ if $$n$$ is even
B.
0 if $$n$$ is odd
C.
0 if $$n$$ is even
D.
$$\pi $$ for all $$n\, \in \,N$$
Answer :
0 if $$n$$ is even
Solution :
$$\eqalign{
& I = \int_0^\pi {\frac{{\sin \,n\left( {\pi - x} \right)}}{{\sin \left( {\pi - x} \right)}}dx = \int_0^\pi {\frac{{{{\left( { - 1} \right)}^{n - 1}}\sin \,nx}}{{\sin \,x}}dx} = {{\left( { - 1} \right)}^{n - 1}}I} \cr
& \therefore {\text{ if }}n\,{\text{is even, }}I = - I\,\, \Rightarrow I = 0 \cr} $$