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Integration by Parts

Q:  ∫e^(2x) * sin(3x) dx


A:  This is a more involved, integration by parts problem.  So, I will assume you know the basics of integration.  If you need more explanation on any step, let me know!

Integration by Parts (the gist):  ∫udv = uv – ∫vdu

So, I always fill in the following chart:

u =                                v =

du =                              dv =

Remember, we get to “select” the u and the dv.  The v and the du are found via differentiation or integration.  So, I select the following

u = e^(2x)                    v =

du =                              dv = sin(3x) dx

Now, fill in the rest:

u = e^(2x)                    v = – 1/3 cos(3x)

du = 2e^(2x) dx          dv = sin(3x) dx

So, we know that the solution to our original problem is:

∫udv = uv – ∫vdu

And, I plug in the parts to get (with a little house-keeping):

(1) ∫e^(2x) * sin(3x) dx = – 1/3 e^(2x) * cos(3x) + 2/3 ∫e^(2x) * cos(3x) dx

Hmmm.  Now we have another integration to do (by parts even)!  We will select the “e” part to be u again, and the “cos” part to be dv.  This way, we do not undo what we just did.  So, just removing the integration above for our analysis, we have:

(2) ∫e^(2x) * cos(3x) dx

u = e^(2x)                    v =  1/3 sin(3x)

du = 2e^(2x) dx          dv = cos(3x) dx

∫e^(2x) * cos(3x) dx = 1 / 3 e^(2x) * sin(3x) – 2/3 ∫e^(2x) *sin(3x) dx

Notice that the answer above contains the original problem?!?  This is great news.  This is what I call a “cycling” problem.  Watch below:

So, going back to equation (1):

∫e^(2x) * sin(3x) dx = – 1/3 e^(2x) * cos(3x) + 2/3 ∫e^(2x) * cos(3x) dx

Plug in our solution for (2):

∫e^(2x) * sin(3x) dx = – 1/3 e^(2x) * cos(3x) + 2/3 [1 / 3 e^(2x) * sin(3x) – 2/3 ∫e^(2x) *sin(3x)]

Clean house:

∫e^(2x) * sin(3x) dx = – 1/3 e^(2x) * cos(3x) + 2/9 e^(2x) * sin(3x) – 4/9 ∫e^(2x) *sin(3x) dx

Add [4/9 ∫e^(2x) *sin(3x) dx] to each side of the equation to get:

13/9 ∫e^(2x) *sin(3x) = – 1/3 e^(2x) * cos(3x) + 2/9 e^(2x) * sin(3x) dx

Multiply both sides by 9/13:

∫e^(2x) *sin(3x) dx = – 9/39 e^(2x) * cos(3x) + 2/13 e^(2x) * sin(3x)

Tada!

3 thoughts on “Integration by Parts

  1. Thank you very much the explanation makes it so easy!

  2. Glad it was helpful! These cycling problems can be messy and require good “house keeping”.

  3. you aint kiddin they do! Had a couple of these for homework and now that you showed me this one I can get through the mess of the other ones lol

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