3.1 Integration by Parts

[STARTS AT 5:05]

Professor: Let's try this again. We have the integral of x, e to the 2x. Now u substitution actually works fairly nicely here. Oh, no it doesn't, because if I take the derivative of 2x I just get 2, there's no x in there. 

OK. So let's try integration by parts where u is x-- I'm going to write this in a table form-- u and dv. So u is x and dv, we'll use e to the 2x. Now the derivative of x is 1. The integral-- I'm integrating down this way and I am differentiating down this way. The integral of that will be 1/2 e to the 2x. 

Now that means my integral is going to be u times v. If you notice, that is going to be u times v. So 1/2x e to the 2x minus the integral of vdu. That is v right here times du. We're multiplying those two. 

So it equals the integral of 1/2e to the 2x dx. Well, u substitution will work very well right here. So this second interval is minus 1/4e to the 2x plus c. Then I have 1/2x e to the 2x. 

And that's all we have. It's not so much a trick. It's a technique that we can apply and it doesn't work all the time. We have to be kind of careful. The one thing that we want to make sure of is that u is a function we can easily differentiate and v is a function that's easily integrable. Keep that in mind.

[ENDS AT 7:10]