Calculus – How to Integrate When Integration by Parts Never Ends

calculusintegration

So I have the following integration: $\int{e^{-x}\sin{x}}$

I let u be the $e^{-x}$ and $dv = \sin{x} dx$

After doing the integration by parts I get:$(e^{-x})(-\cos{x})-\int{-e^{-x}*(-\cos{x})}$

From what it looks like, if I continue integration by parts, will it just continue going on forever?

Best Answer

For integration by parts, you will need to do it twice to get the same integral that you started with. When that happens, you substitute it for L, M, or some other letter. So we start by taking your original integral and begin the process as shown below.

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Now using the formula

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We get the following sequence, and we do another round of integration by parts.

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So we end up the integral part that is the same as your original equation like so:

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Now we substitute your original integral with the letter L like so...

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...and replace every occurrence so the equation looks like the following:

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Then we solve for L as shown below.

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So your complete solution is this:

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Hope this helps.

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