Integration by parts
Integration by parts: definite integrals
- [Instructor] We're gonna do in this video is try to evaluate the definite integral from zero to pi of x cosine of x dx. Like always, pause this video and see if you can evaluate it yourself. Well when you immediately look at this, it's not obvious how you just straight up take the anti-derivative here and then evaluate that at pi and then subtract from that and evaluate it at zero, so we're probably going to have to use a slightly more sophisticated technique. And in general, if you see a product of functions right over here, and if one of these functions is fairly straightforward to take the anti-derivative of without making it more complicated like cosine of x, and another of the functions like x if you were to take its derivative, it gets simpler. In this case, it would just become one. It's a pretty good sign that we should be using integration by parts. So let's just remind ourselves about integration by parts. So integration by parts, I'll do it right over here, if I have the integral and I'll just write this as an indefinite integral but here we wanna take the indefinite integral and then evaluate it at pi and evaluate it at zero, so if I have f of x times g prime of x, dx, this is going to be equal to, and in other videos we prove this, it really just comes straight out of the product rule that you learned in differential calculus, this is gonna be equal to f of x times g of x minus, you then swap these around, minus f prime of x, g of x, dx. And just to reiterate what I said before, you wanna find an f of x that when I take its derivative, it simplifies it, so simplify, and you wanna find a g prime of x that when I take its anti-derivative, so when I take its anti-derivative, it doesn't get more complicated. So not more complicated. Because if the f of x gets simplified when I take its derivative, and the g prime of x does not get more complicated when I take its anti-derivative, then this expression will maybe be easier to find the anti-derivative of. So let's do that over here. Between x and cosine of x, which one gets more simple when I take its derivative? Well the derivative of x is just one, so I'm gonna make that my f of x, so I could write that over here, so my f of x, I will say is x, in which case, f prime of x, f prime of x, is going to be equal to one, and then what would my g prime of x be? Well my g prime of x, cosine of x if I take its anti-derivative, it doesn't get more complicated. The anti-derivative of cosine of x is sine of x. So let me make that my g prime of x. So g prime of x is equal to cosine of x in which case g of x, the anti-derivative of cosine of x, well it's just sine of x or another way to think about it, the derivative of sine of x is cosine of x. Now you could think about plus C's and all of that but remember, this is gonna be a definite integral so all of those arbitrary constants are going to get canceled out. So now let's think through this. Let's just apply the integration by parts here. In this particular case, all of this is going to be equal to, so I'm saying that is equal to this, I'm gonna skip down here, it's going to be equal to f of x times g of x. So that is f of x is x, g of x is sine of x, f of x times g of x, minus the integral of f prime of x, f prime of x is just one, we could write it like that, one times g of x, g of x is sine of x, so we could write it like this, but one times sine of x, well we could just rewrite that as sine of x, it'll make it a little bit simpler, sine of x, dx, and then remember, this is a definite integral, so we are going to want to evaluate this whole thing at pi and at zero, and then take the difference between the two. But what is the indefinite integral of sine of x dx? Well, or to say the anti-derivative of it, we know that the derivative of cosine is negative sine of x, and so in fact what we want, we could bring this negative sine into the integral, so we could say plus the integral of negative sine of x, now this clearly the anti-derivative here is cosine of x, so this thing is going to be cosine of x, and now we just have to evaluate it at the end points. So let's first evaluate this whole thing at pi. So this is going to be equal to pi sine of pi, pi sine of pi, plus cosine of pi, and then from that I'm going to subtract this whole thing evaluated at zero, so let me do zero in a different color, at zero, so it's going to be zero times sine of zero plus cosine of zero, so let's see, sine of pi is just zero so this is just going to cancel out. Cosine of pi, that is negative one, and then this is zero, and then cosine of zero, that is one, so you have negative one minus one, so this all gets us to negative two, and we are done. Using integration by parts, we were able to evaluate this definite integral.