𝘶-substitution: double substitution
Let's see if we can take the integral of cosine of 5x over e to the sine of 5x dx. And there's a crow squawking outside of my window, so I'll try to stay focused. So let's think about whether u-substitution might be appropriate. Your first temptation might have said, hey, maybe we let u equal sine of 5x. And if u is equal to sine of 5x, we have something that's pretty close to du up here. Let's verify that. So du could be equal to-- so du dx, derivative of u with respect to x. Well, we just use the chain rule. Derivative of 5x is 5. Times the derivative of sine of 5x with respect to 5x, that's just going to be cosine of 5x. If we want to write this into differential form, which is useful when we do our u-substitution, we could say that du is equal to 5 cosine 5x. And we look over here, we don't have quite du there. We have just cosine of 5x dx. Cosine of 5x dx just like that. So when you look over here, you have a cosine of 5x dx, but we don't have a 5 cosine of 5x dx. But we know how to solve that. We can multiply by 5 and divide by 5. 1/5 times 5 is just going to be 1, so we haven't changed the value of the expression. When we do it this way, we see pretty clearly we have our u and we have our du. Our du is 5. Let me circle that. Let me do that in that blue color-- is 5 cosine of 5x dx. So we can rewrite this entire expression as-- do that 1/5 in purple. This is going to be equal to 1/5-- I hope you don't hear that crow outside. He's getting quite obnoxious. 1/5 times the integral of all the stuff in blue is my du, and then that is over e to the u. So how do we take the antiderivative of this? Well, you might be tempted to well, what would you do here? Well, we're still not quite ready to simply take the antiderivative here. If I were to rewrite this, I could rewrite this as this is equal to 1/5 times the integral of e to the negative u du. And so what might jump out at you is maybe we do another substitution. We've already used the letter u, so maybe we'll use w. We'll do some w-substitution. And you might be able to do this in your head, but we'll do w-substitution just to make it a little bit clearer. This would have been really useful if this was just e to the u because we know the antiderivative of e to the u is just e to the u. So let's try to get it in terms of the form of e to the something and not e to the negative something. So let's set. And I'm running out of colors here. Let's set w as equal to negative u. In that case, then dw, derivative of w with respect to u, is negative 1. Or if we were to write that statement in differential form, dw is equal to du times negative 1 is negative du. So this right over here would be our w. And do we have a dw here? Well, we have just a du. We don't have a negative du there. But we can create a negative du by multiplying this inside by a negative 1, but then also by multiplying the outside by a negative 1. Negative 1 times negative 1 is positive 1. We haven't changed the value. We have to do both of these in order for it to make sense. Or I could do it like this. So negative 1 over here and a negative 1 right over there. And if we do it in that form, then this negative 1 times du, that's the same thing as negative du. This is this right over here. And so we can rewrite our integral. It's going to be equal to now it's going to be negative 1/5. Trying to use the colors as best as I can. Negative 1/5 times the indefinite integral of e to the-- well, instead of negative u, we can write w-- e to the w. And instead of du times negative 1, or negative du, we can write dw. Now this simplifies things a good bit. We know what the antiderivative of this is in terms of w. This is going to be equal to negative 1/5 e to the w. And then we might have some constant there, so I'll just do a plus c. And now we just have to do all of our unsubstituting. So we know that w is equal to negative u. So we could write that. So this is equal to negative 1/5. I want to stay true to my colors. Negative 1/5 e to the negative u, that's what w is equal to, plus c. But we're still not done unsubstituting. We know that u is equal to sine of 5x. u is equal to sine of 5x, so we can write this as being equal to negative 1/5 times e to the negative u, which is negative u is sine of 5x. And then finally, we have our plus c. Now, there was a simpler way that we could have done this by just doing one substitution. But then you would have had to look ahead a little bit and realize that it was not trivial to take your-- not too bad to take your antiderivative of e to the negative u. The insight that you might have had-- although you shouldn't really hold yourself, feel too bad if you didn't see that insight-- is that we could have rewritten that original integral-- and let me rewrite it. It's cosine of 5x over e to the sine of 5x dx. We could have written this entire original integral as being equal to cosine of 5x times e to the negative sine of 5x dx. And in this situation, we could have set u to be equal to negative 5x and say, well, if u is equal to negative sine of 5x. If u is equal to negative sine of 5x, then du is going to be equal to negative 5 cosine of 5x. And we don't have a negative-- oh, dx. We don't have a negative 5 here, but we could construct one by putting a negative 5 there and then multiplying by negative 1/5. And then that would have immediately simplified this integral right over here to be equal to negative 1/5 times the integral of well, we have our du. Let me do this in a different color. We have our du. That's the negative 5. Let me do it this way. Negative 5 cosine of 5x dx. So that is our du. I'm just changing the order of multiplication. Times e to the u. This whole thing now is u this second time around. So if we did it this way, with just one substitution, we could have immediately gotten to the result that we wanted. You take the antiderivative of this. I'll do it in one color now just because I think you get the idea. This is equal to negative 1/5 e to the u plus c. u is equal to negative sine of 5x. So this is equal to negative 1/5 e to the negative sine of 5x plus c And you're done. So this one is faster. It's simpler. And over time, you might even start being able to do this in your head. This top one, you still didn't mess up by just setting u equal to sine of 5x, we just have to do an extra substitution in order to work it through all the way. And I was able to do this video despite the crowing crow outside, or squawking crow.