Integral sin(3x pi/a)sin(2x pi/a)cos(x pi/a)

I'm doing a bit of quantum mechanics and I'm doing a question on the infinite square well.

Anyway, I'm a bit stuck on this integral:

$\displaystyle \displaystyle \sqrt{\dfrac{2}{a}} \dfrac{2}{\sqrt{a}} \int_0^a \sin \left( \dfrac{3 \pi}{a} x\right) \sin \left( \dfrac{2 \pi}{a} x\right) \cos \left( \dfrac{\pi}{a} x\right)dx$

Does anyone know how to integrate this?

I tried reversing the product rule for 3 functions and got this: $\displaystyle \displaystyle \int u v w' = u v w - \int u' v w - \int u v' w $

But I didn't really get anywhere.

I know that these functions are orthogonal to each other (except for the cosine) and so if they were all sines I can use the Kronecker Delta function to evaluate them faster, but that cosine really messes things up.

Using Wolfram Alpha I got $\displaystyle \dfrac{\sqrt{2}}{2}$ but I want to know how to do this.

Re: Integral sin(3x pi/a)sin(2x pi/a)cos(x pi/a)

remember the sum and product formula for trig functions.

2 sinAsinB = cos(A-B) - cos ( A+B)

2 cosAcosB = cos(A+B) + cos ( A-B)

2 sinAcosB = sin(A+B) + sin ( A-B)

2 cosAcosB = sin(A+B) - sin ( A-B)

Apply the appropriate one and you will have your integral reducing to simple integral of trig ratios