Question

If $m$ and $n$ are integers such that $m+n$ is an even integer,

then the value of ${\int }_0^{\pi }\cos mx\sin nxdx$ is?

• A. $\frac{\pi }{2}$
• B. $\frac{2n}{n^2-m^2}$
• C. $0$
• D. $\frac{2mn}{m^2-n^2}$

Answer 1

The correct option is C $0$

$I={\int }_0^{\pi }\cos mx\sin nxdx$

Using ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$

$I={\int }_0^{\pi }\cos (\pi -mx)\sin (\pi -nx)dx=-I\therefore 2I=0\Rightarrow I=0$