Fourier sine series for function F(t) = t for 0<t<L

Question

How to get to the part circled with red? I tried to compute it on Wolfram alpha.. (http://www.wolframalpha.com/input/?i=2%2FL+*Integrate+x+sin%28%28n+pi+x%29%2FL%29+dx+from+0+to+L) Still confused. Need help.

Answer 1

It's all right, the only problem is that Wolfram Alpha didn't know that $n$ is an integer. For a non-integral $n$, you can't in general simplify Wolfram Alpha's result. For $n$ an integer, you can simplify

$$\sin (n\pi) = 0$$

and

$$\cos (n\pi) = (-1)^n$$

to get the simpler result.