# Help with evaluating an integral

As part of a problem, I'm stuck evaluating the following integral

$$\int_0^\pi \sin\left(\left|\frac{x}{2}-\frac{y}{2}\right|\right) \sin (2y)~dy.$$ I seek assistance in evaluating it. Thank you very much.

• Is there any interval defined for x and y? – Ram Jan 23 '13 at 5:43
• no Ram. there is no interval. – Paul Jan 23 '13 at 5:59
• I mean is there any relation between x and y, I hope x is like $0 \le x \le \pi$ then you can reduce into two intervals, $sin(x/2 - y/2)$ in $0 \le y \le x$ and $sin(y/2 - x/2)$ in $x < y \le \pi$ – Ram Jan 23 '13 at 6:16
• @Ram: yes $0\le x \le \pi$. Thanks. – Paul Jan 23 '13 at 6:24

First, break it into two pieces, one from $0$ to $x$, the other from $x$ to $\pi$. This way, you can write an equivalent integral without absolute value signs.
Then can you do $\int\sin(a-by)\sin(cy)\,dy$? It might help to remember there's a trig identity for $\sin A\sin B$.
• Would this work: $$\int_0^x \sin(x-y)\sin(2y)-\int_x^\pi sin(x-y)\sin(2y) dy?$$ – Paul Jan 23 '13 at 5:58
• It appears you've misplaced the $2$s in the denominators but, aside from that, it looks like what I had in mind. – Gerry Myerson Jan 23 '13 at 6:11