Integration Question: How to Solve e^8x * sin(x) dx with Homework Equations

[alacey11]

Homework Statement

int e^8x * sin(x) dx

Homework Equations

I can integrate each of them separately - it's the multiplication that confuses me.
Is there some sort of product rule for integration?
I'm not sure where to start, I just need a push in the right direction.

The Attempt at a Solution

This is part of a larger problem, but the rest is irrelevant.
Thanks

 

[theJorge551]

There is a product rule, per say, for integration. It's pretty easy to derive, all you have to do is write out the product rule for differentiation, flip the operations from dy/dx to ∫ f(x) dx, and you can pretty quickly come to a conclusion by rearranging the equation.

 

[alacey11]

I think I'm doing it wrong, because I just got two integrals that were just as hard:

int (e^8x * cos(x) dx) + int ((e^8)/8 * sin(x) dx)

 

[Mark44]

The integration counterpart to the product rule in differentiation is called integration by parts, and that's probably what theJorge551 was alluding to.

If you do integration by parts twice, and have chosen the parts carefully, you will get an equation that you can solve algebraically for
$$\int e^{8x} sin(x)dx$$

 

[theJorge551]

Thank you for clarifying, Mark; that is what I was alluding to.

 

[alacey11]

I have solved it now...
I was familiar with the integration by parts, but would not have thought to use it twice - I had used it once and when I saw the new integral with cos() I assumed I had done it wrong.
Thanks a lot!