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

[AxiomOfChoice]

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

[tex]
\int_0^T \sin(at+b) \cos(ct+d) dt
[/tex]

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

Thanks!

 

[Ben Niehoff]

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

 

[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.

 

[wsalem]

I would choose the exponential form which makes it a matter of algebraic manipulation, but using trigonometric identities is equality valid.
[tex]sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}[/tex]
and [tex]cos(\theta) = = \frac{e^{i\theta} + e^{-i\theta}}{2}[/tex]
substituting in the original equation, you'll eventually end up with an expression of the form
[tex]\frac{e^{ix} - e^{-ix} + e^{iy} - e^{-iy}}{4i}[/tex]
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)

 

[arildno]

Remember that:
[tex]\sin(u)\cos(v)=\frac{1}{2}(\sin(u+v)+\sin(u-v))[/tex]

 

[AxiomOfChoice]

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

Thanks though.