 Let's do a quick review of the main ideas of section 5.3 in active calculus on integration by substitution. The first fundamental theorem of calculus told us in a previous section that to compute a definite integral, one way to go about it is to find an antiderivative of the integrand, and then simply plug in the upper and lower limits of integration and subtract. So what we are embarking upon in this section is a thorough basic study of techniques for calculating antiderivatives. Recall that we just used the integral sign by itself with no limits attached to represent the most general antiderivative of a function, and we sometimes call that an indefinite integral. Some antiderivatives are straightforward. For example, those that involve basic polynomials. For example, the indefinite integral of x cubed plus 2x plus 1 is 1 fourth x to the fourth plus x squared plus x plus a constant c. But for other functions, the rules for anti-differentiation aren't as simple. For instance, what is the indefinite integral of sine of 4x? We might be tempted to say it's negative cosine of 4x because an antiderivative of sine of only x is negative cosine of x. But negative cosine 4x is not correct because this is not an antiderivative. The derivative of negative cosine 4x is not sine of 4x, but rather 4 sine of 4x. So we're off, but notice we're not off by much, only by that factor of 4. That factor of 4 showed up because in checking to see whether the derivative of our answer actually does equal the original integrand, sine 4x, we had to use the chain rule, and the chain rule introduced that extra factor of 4. So what this should tell you is that integrals, whether they're definite or indefinite, that involve undoing the chain rule somehow, will require a little extra thinking, and the technique that embodies and encodes that thinking is known as the method of substitution. The general idea of substitution is that we're first going to identify somewhere in the integrand a composite function with an inside and an outside. The inside function is what causes that imbalance or extra factors when we try to integrate like the 4 in our example a minute ago. So in the method of substitution, we're going to substitute out this problematic inside function by assigning it to a new variable name. Often we use the variable u, and so sometimes you hear this method called u substitution. Once we do this, there's going to no longer be a complicated inside function, which is great for us, but there's a price to pay. We now have to continue to replace all instances of the original variable with that new variable u. This means that we have to do some additional algebra to convert over any expressions in x that show up in the integrand into expressions that involve only the new variable u, and that includes the differential. We'll explore how to do this in various test cases in the next few videos. For now, it suffices to say that once all these substitutions have taken place, and if all the algebra is done correctly, what we should be left with is a new integral that is similar to the previous one, but simpler, one that can be computed directly. And then we compute that integral directly, and at the end of the computation, we just need to reverse the substitution and get back to the original variable. That's the broad strategy behind integration by substitution. Now let's see how it works in practice.