Finding integrals of the product of trig functions

[DrWillVKN]

Homework Statement

I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function.

One of them is \intsin(a)*sin(b - a)da
Another is \inte(a)*sin(a)da

Homework Equations

sin(x + y) =sinx(cosy) + siny(cosx)
cos(x + y) = cosx(cosy) - sinx(siny)
sin^2(x) = (1 - cos2x)/2

The Attempt at a Solution

I've tried to use the trig identity for sin(b-a), but that just gives an extremely long sin and cos statement that doesn't help. the one with e is even more confusing. How am I supposed to manually solve them?

 

[Char. Limit]

For the second problem, let

I = \int e^{a} sin(a) da

Then integrate the right side by parts twice, and note that you get, as your integral, I. From there, just solve for I.

 

[LCKurtz]

Another trick for integrating integrals like ∫e^xsin(x)dx is to instead do the integral ∫e^xe^ixdx =∫e^(1+i)xdx as a simple exponential. Rationalize the result and note your original integral is the imaginary part. This avoids two integrations by parts and gives you the ∫e^xcos(x)dx from the real part as a free bonus.

 

[DrWillVKN]

thanks for the replies, found the answers. i just didn't go far enough with the identities and integration by parts.