**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!

Thank you very much the explanation makes it so easy!

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

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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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