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

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SUMMARY

The integral of (eax)sin(bx) can be solved using integration by parts, requiring two applications of the method. The representation of sin(bx) in terms of exponential functions, specifically sin(bx) = (eibx - e-ibx)/2i, simplifies the process. After applying integration by parts twice, the original integral reappears as a multiple, allowing for isolation and solution. This approach clarifies a common misunderstanding among students who typically apply integration by parts only once.

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How do I integrate (e^ax)sin(bx)
 
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I think integration by parts will work.
 
darthxepher said:
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.
 
danago said:
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.
 

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