Posted by **Emma** on Wednesday, May 18, 2011 at 2:14am.

Please help me solve this integration problem. I don't understand how to do it. Thank you!

∫e^(2x)*sin(2x)dx

- Calculus - Integration by Parts -
**drwls**, Wednesday, May 18, 2011 at 7:44am
I do not believe you can do this in one integration-by-parts step. You must use the method twice.

Let y = 2x and dx = dy/2, to simplify the problem to

(1/2)∫e^y*sin(y)dy

Next, let e^y = u and dv = siny dy

du = e^y dy and v = -cos y

(1/2)∫e^y*sin(y)dy = (1/2)*uv -(1/2)∫v*du

= (1/2)[-u cos y +∫e^y cosy dy]

Now you must apply integration by parts a second time on the

+∫e^y cosy dy term

This will give you an equation for

∫e^y*sin(y)dy that involves explicit functions of y.

When you are all done, substitute 2x for y

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