|
|
a)
$I=\int\limits_{-\pi}^{\pi}\sin mx\sin nxdx$
$=\int\limits_{-\pi}^{\pi} \frac{1}{2}(-\cos(m+n)x+\cos(m-n)x)dx$
+ Nếu $m=n$
$I=\int\limits_{-\pi}^{\pi} \frac{1}{2}(-\cos2mx+1)dx=\left[ {- \frac{1}{4}\sin2mx+\frac{1}{2}x} \right]_{-\pi}^{\pi}=\pi$
+ Nếu $m=-n$
$I=\int\limits_{-\pi}^{\pi} \frac{1}{2}(\cos2mx-1)dx=\left[ { \frac{1}{4}\cos2mx-\frac{1}{2}x} \right]_{-\pi}^{\pi}=-\pi$
+ Nếu $m\ne \pm n$
$I=\frac{1}{2}(\frac{-\sin(m+n)x}{m+n}+\frac{\sin(m-n)x}{m-n})dx\left|\begin{array}{l}\pi\\-\pi\end{array}\right.=\frac{\sin(m+n)\pi}{m+n}+\frac{\sin(m-n)\pi}{m-n}$
|