 Hello and welcome to Screencast about integration by substitution. The example we're going to look at today is an indefinite integral. So it's the integral of sine of 4x dx and whenever you kind of use those words of or whenever you think of, oh, this would be a chain rule problem if I were doing the derivative, any kind of a composite type function, that's how you know you're going to want to use your substitution. Okay, so then we want to be able to form our function derivative pair and let me get a pen color here. So the way you want to do that is again kind of look for your inside function, look for the inner function, look for something that when you take the derivative of it, you can match it up with something else that's in the integral. Okay, so for example for this one, our inside function is 4x because that's the function inside of that sine function. And when I go to do the derivative of that function, that gives me 4, dx. Okay, so this is your Leibniz notation and that kind of stuff, so don't worry too much about that, but just make sure that you've got all your pieces in here that we need to match up. Okay, so sine, we know how to do the anti-derivative or the integral of sine. We've got our function here, so this is going to become our U. So now we have to match up this dx. Well, you notice when we did our derivative we had 4dx. So how can I get rid of that 4? Well, divide by 4 or multiply by 1 fourth. So 1 fourth du is dx. Okay, so now I'm going to rewrite my integral. So that's going to come over here. It's going to be the sine of U because that's what my 4x is. And then I've got 1 fourth of du because that's what my dx is. So just kind of think of this as a puzzle and you're just trying to fit all the pieces together. Now having this 1 fourth in between the sine function and the du is a little bit awkward. So hopefully you guys remember that you can just pull those constants right outside of the integral. So we now have 1 fourth the integral of sine U du. Well, sine of U, that's a function I can definitely find the anti-derivative of. So remember to do that, you want to ask yourself the question, what function, when I take the derivative of it, will give me sine of U? And that of course is negative cosine. So my 1 fourth needs to come along. I've got negative cosine U and then of course since this is an indefinite integral, we always want to put our plus c on there to generalize this and know that we've got some constant hanging around. So now all we need to do now is just take our U and plug it back in. So our final answer here is going to be negative 1 fourth because I like those two things to go together a little bit better. Cosine of 4x plus c. So that would be then your indefinite integral for this. How do you check it? You take the derivative. So I'll just take that for just, we'll just do this one time. So to check it then I want to do the derivative, so my negative 1 fourth is going to come along because it's a constant. The derivative of cosine is negative sine. The inside function stays the same with the chain rule. And then I have to multiply by the derivative of that inside function, which is 4. And guess what? We get right back to where we started because the negatives are going to cancel each other out, the fours are going to cancel each other out, so I end up with the sine of 4x, which is exactly what I wanted. Thank you for watching.