How Do You Solve the Integral of (e^ax)sin(bx) Using Integration by Parts?

[darthxepher]

How do I integrate (e^ax)sin(bx)

 

[danago]

I think integration by parts will work.

 

[mathman]

How do I integrate (e^ax)sin(bx)

A somewhat clumsy, but direct, method is to use the representation of sin(bx) in terms of exp. Specifically:

sin(bx)=(e^ibx - e^-ibx)/2i

This gives you two integrals of the exp function. You can then do a little playing around to get rid of the i terms.

 

[rs1n]

I think integration by parts will work.

Integration by parts will work, but there's a bit more to it. You will need to apply integration by parts twice. After the second application, a multiple of your original integral will reappear. You would then have to isolate this integral. In other words, you will obtain something like:

I = (stuff from using integration by parts twice) + a*I

where 'I' represents your original integral and 'a' is some constant. You'd then have to "solve" for the 'I'

I just thought I'd add this because oftentimes students only apply integration by parts once and do not see the solution right away, and get flustered.