 Let's do a quick review of section 5.4 in active calculus on integration by parts. We're continuing to study computational techniques for finding antiderivatives of functions. So far we've seen a couple of general methods. The first method is to simply use basic facts about derivatives to construct antiderivatives, which we can then use to calculate definite integrals. For example, we can use the power rule in reverse to find an antiderivative for x cubed plus 2x plus 1, or the rule about the derivative of arc tangent to find an antiderivative of 5 over x squared plus 1. We also just learned the substitution method, which allows us to find more complex antiderivatives, usually when we are integrating functions that are the result of the chain rule, and have a composite function structure. For example, we can use the substitution method to find an antiderivative of cosine of 5x or 5 over 100x squared plus 1 by defining the right value of that variable u, and then making a substitution that creates a simpler integral in the end. This section studies a very important method for integration called integration by parts. Integration by parts is similar to the method of substitution, in that it involves making substitutions to group a piece of an integrand into a new variable. And the substitution, once fully worked out, creates a much simpler integral. But integration by parts is more complex than basic u substitution, and the process of choosing and making the substitution has more algebra involved. Whereas the method of substitution comes from the chain rule, integration by parts comes from the product rule. And so this method is often used on integrals where the integrand is a product of two functions. So let's first recall what the product rule says. It says that the derivative of the product of two differentiable functions, f and g, is f prime of x times g of x plus f of x times g prime of x. If we take that and integrate both sides of this equation with respect to x, we get f of x times g of x equals the integral of f prime times g dx, plus the integral of f g prime dx. If we subtract the first integral to the other side, we get this equation which says that the integral of f of x g prime of x dx is equal to f of x times g of x with no integral on it, minus the integral of f prime times g dx. In integration by parts, what we want to do is start with an integral that contains a product of two functions, and then identify one of those two functions as a derivative. That is, one of those two functions involved should be easy to anti-differentiate. We often write u and dv instead of f of x and g prime of x dx, respectively. If we can identify one of the two functions in the product as being easy to differentiate, and the other as being easy to anti-differentiate, then the formula known as the integration by parts formula allows us to rewrite the integral in terms of a product minus another integral. This at first seems like no progress because we still have an integral to evaluate, but the idea is that the second integral should be easier to compute than the original one. So on the left, we have a hard integral, and on the right, if we go through the process of separating the integral out into its parts and making correct substitutions, we'll have a basic product that involves no calculus and a new integral that's easier to work with. That's the overall strategy of integration by parts. Now let's see several examples of how this works in practice.