Integration question verifying piecewise

Question

I have the following question: from direct integration show

$\displaystyle \int \limits_{-L}^{L} \cos({m πx\over L})\cos({nπx\over L}) \ dx = \begin{cases}0 & m \neq n \\ L & m = n \\ \end{cases} $

I use a trigonometric identity and evaluate individually to produce:

$\displaystyle {L\sin(π(m-n)\over πm-πn} + {L\sin(π(m+n)\over πm+πn}$

I am pretty sure I am correct but I can quite clearly see that whenever m=n the part on the left will not exist let alone equal L. Any hints would be appreciated.

Cheers

Answer 1

It's safest to consider the case $m=n$ separately when solving the integral. But also note that $\lim_{x\rightarrow 0}\frac{\sin x}{x}=1$.