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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)