Integrate xe^3x

∫ xe^3x dx: This is a simple integration to perform and ideal if you are starting out.

For the u substitution, I am choosing x, and its derivative is 1 as you can see

I am choosing dv/dx to be e^3x, and therefore v is 1/3 e^3x.

At this stage, we can substitute the values of u, v, and du/dx into the standard integration by parts formula to give the expression shown above. Everything is exactly in its original place so that you can follow easily. As you can see, we have one term on the RHS that still requires integration.

We are close to the result.

Finally, a small amount of tidying up gives us this final answer for presentation. You could factorise out the 1/3 e^3x but I prefer it this way.