 Hello, and welcome to the Grand Valley State University Calculus screencasts. Today we're going to talk about differentiating integral functions. We've seen that we can use the definite integral to define functions by allowing the upper limit to be a variable. And whenever we differentiate an integral function like that, the result is the integrand, and this is what we call the second fundamental theorem of calculus. And the second fundamental theorem of calculus is fundamental in the sense that it shows that any continuous function has an anti-derivative, even if we can't find a nice formula for it. In particular, the definite integral from a to x of f of t dt is an anti-derivative for the integrand f. Put another way, the second fundamental theorem of calculus says that differentiation undoes integration. Now in this screencast, we're going to use the second fundamental theorem of calculus, along with other properties of integrals and properties of the derivative in order to calculate derivatives of three specific functions defined by definite integrals. So for our first example, consider the function capital F defined as the definite integral from 1 to x of cosine t squared plus 1 dt. Now this is an example to which we can directly apply the second fundamental theorem of calculus to find the derivative big f prime. So go ahead and pause the video for a moment and see if you can figure out how the second fundamental theorem of calculus applies here, then resume when you're ready. Recall that the second fundamental theorem of calculus tells us that differentiation undoes integration. In other words, if we integrate and then differentiate, we should just get back the integrand as a function of x. So the derivative of big f of x in this case is just the cosine of x squared plus 1. We take the integrand and evaluate it at x. Now we can use the second fundamental theorem of calculus and some properties of the definite integral to differentiate slightly more complicated functions. For example, let a of x be the definite integral from x to 3 of e to the tangent of t dt. Now note that this function capital A doesn't have the form integral a to x of f of t dt that's in the second fundamental theorem of calculus since the variable limit in a is the lower limit and not the upper limit. So we can't directly apply the second fundamental theorem of calculus to find a prime. However, we can use properties of the definite integral to rewrite a so that it's in a slightly different form to which we can apply the second fundamental theorem of calculus. Pause the video for a moment and see if you can figure out how to do that and then calculate the derivative of big a, and then resume the video when you're ready. Recall that one of the properties of the definite integral is that we can reverse the limits of integration by multiplying the integral by negative 1. This will allow us to rewrite capital A so that the upper limit is the one that has the variable. In other words, capital A of x can be written as the opposite of the definite integral from 3 to x of e to the tangent t dt. And now that the variable limit is the upper limit, we can apply the second fundamental theorem of calculus to find the derivative of capital A. Since we can factor out the constant multiple negative 1 from the derivative, we see that big a prime of x is the opposite of the derivative of the integral from 3 to x of e to the tangent t dt. The second fundamental theorem of calculus tells us that that's just the opposite of the integrand evaluated of x. So a prime of x is the opposite of e to the tangent of x. We can also combine the second fundamental theorem of calculus with some of the differentiation rules we know to differentiate even more complicated functions defined by integrals. For example, suppose we take capital T of x, which is the integral from 1 to the natural log of x, of 1 over 1 plus t to the fourth dt. Note that T of x doesn't have the form integral 8x of f of t dt as we need to use the second fundamental theorem of calculus because the upper variable limit isn't just x, it's a function of x. So we can't directly apply the second fundamental theorem of calculus to determine the derivative of capital T. However, if we're clever, we can view capital T as a composite of functions and then use both the second fundamental theorem of calculus and the chain rule to differentiate t. Pause the video for a moment and think about how we might write capital T of x as a composite big f of g of x and clearly identify what big f of x and what g of x are. You might want to think about how we would relate big t to the integral from 1 to x of 1 over 1 plus t to the fourth dt so that we can use the second fundamental theorem. So resume the video when you're ready. Okay, we want to write big t as a composite. Let big f of x be the definite integral from 1 to x of 1 over 1 plus t to the fourth dt. Then to get big t, we want to replace that x in that upper limit with ln x. So we might just let g of x be ln of x. If we do that, then big t of x is in fact big f of g of x. Now that we have big t written as a composite, we can use the chain rule to differentiate t. Note that the derivative of big t of x is going to be the derivative of big f of g of x so it will be f prime of g of x times g prime of x. Now the second fundamental theorem of calculus tells us how to differentiate big f. Remember big f of x is the integral from 1 to x of 1 over 1 plus t to the fourth dt. So the derivative of big f with respect to x is just the integrand as a function of x, 1 over 1 plus x to the fourth. That means that f prime of g of x, we get by evaluating f prime of x at g of x. So we take g of x and substitute it in for x in the rule for f prime. And f prime of g of x is then 1 over 1 plus the fourth power of ln x, which we can write as 1 over 1 plus ln to the fourth of x. Recall that g of x is the natural log of x, so g prime of x is 1 over x. This makes t prime of x the derivative from 1 to ln x, 1 over 1 plus t to the fourth dt, which we've seen as f prime of g of x times g prime of x. We get 1 over 1 plus the fourth power of the natural log of x times 1 over x. So in conclusion, we can combine the second fundamental theorem of calculus with rules of differentiation and properties of the definite integral to differentiate lots of functions that are defined by definite integrals. That concludes this screencast, and we hope you'll come back soon.