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

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Discussion Overview

The discussion revolves around the integration of the function (e^ax)sin(bx) using integration by parts. Participants explore different methods and approaches to tackle this integral, including direct application of integration techniques and alternative representations of the sine function.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant inquires about the method to integrate (e^ax)sin(bx).
  • Another participant suggests that integration by parts will be effective for this integral.
  • A third participant proposes using the exponential representation of sin(bx) to transform the integral into two integrals of the exponential function, indicating a more complex approach.
  • A later reply reiterates that integration by parts is applicable but notes that it requires two applications, leading to a reappearance of the original integral, which must be isolated to find the solution.
  • This participant emphasizes that students often overlook the need for multiple applications of integration by parts, which can lead to confusion.

Areas of Agreement / Disagreement

Participants express varying methods to approach the integral, with some agreeing on the utility of integration by parts while others introduce alternative strategies. The discussion does not reach a consensus on a single method or solution.

Contextual Notes

There are indications of missing assumptions regarding the constants involved and the specific steps needed to isolate the integral after applying integration by parts twice. The discussion does not resolve these aspects.

darthxepher
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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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