3.1 Integration by Parts

[STARTS AT 23:39]

Professor: We have x squared sine x. 

OK, now if you notice we actually did the integration by parts formula there twice. So I'm going to go ahead and combine this one, because we're actually going to have to do this one twice. And I see that because I have an x squared. The derivative of x squared is x-- 2x-- and the derivative of that is 2. 

I'm going to combine this in here. So I have dv. I'm going to take take sine x, and this is because this one is going to be cyclic. The integral of sine is negative cosine. The integral of that-- well, let's go ahead and write it out over here. 

This is going to be uv. So that is negative x squared cosine x minus the integral of negative 2x cosine x. If you notice we have the same problem. We have an x term along with our cosine. So that's why we're going to have to apply this again. 

The derivative would be 2 for the next one. And the integral, that would be a negative sine. I'll keep my bracket there. This is going to be u times v-- so negative 2x sine x minus the integral of vdu, which is negative 2 sine xdx. 

Well, we can actually calculate that right there. That is going to be a 2, a negative 2 cosine x. So we got to be careful about our signs there, because-- actually, that'll be a positive. It'll be a positive 2, because of the negative cosine. 

I'm going to take the derivative. And let's bring all of this down. We have negative x squared cosine x plus 2x sine x, distributing that negative. And then we have a positive 2 cosine xdx-- no, dx plus c. And there is our integral.

[ENDS AT 26:15]