Anybody know a general way to approach this sin/cos integral?

1. Apr 2, 2009

AxiomOfChoice

Anyone know a nifty change of variables or trigonometric identity that will make this integral relatively easy to do:

$$\int_0^T \sin(at+b) \cos(ct+d) dt$$

'Cause from where I'm standing, that's pretty awful...

Thanks!

2. Apr 2, 2009

Ben Niehoff

Is the integration over a whole period? That would simplify things significantly.

3. Apr 2, 2009

Georgepowell

well if you use the identities for sin(A+B) and cos(A+B) you can change it into four simpler integrals.

Could you integrate cos(ax)sin(bx) with respect to x? if you can then using the above identities helps.

4. Apr 2, 2009

wsalem

I would choose the exponential form which makes it a matter of algebraic manipulation, but using trigonometric identities is equality valid.
$$sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$$
and $$cos(\theta) = = \frac{e^{i\theta} + e^{-i\theta}}{2}$$
substituting in the original equation, you'll eventually end up with an expression of the form
$$\frac{e^{ix} - e^{-ix} + e^{iy} - e^{-iy}}{4i}$$
getting that back to the trigonometric form, you should end up with two easy integrals, namely the integral of 1/2sin(x) + 1/2sin(y)

Last edited: Apr 2, 2009
5. Apr 2, 2009

arildno

Remember that:
$$\sin(u)\cos(v)=\frac{1}{2}(\sin(u+v)+\sin(u-v))$$

6. Apr 2, 2009

AxiomOfChoice

All of these are great suggestions. But I just wound up doing it by parts. It wasn't too bad.

Thanks though.