Computing Definite Integrals using Substitution - Problem 1
Let’s use the method of substitution to solve another definite integral. Here I’m asked to solve the integral from 1 to e of natural log x over x. First thing I want to do, is rewrite this integral in another form. From 1 to e, I’ll pull the natural log x in front. And this is the same as natural log x times 1 over x dx.
When we substitute, you'll see why I did this. I’m going to substitute for natural log x. Because in here I don’t actually see there is so much as a composite function, but I see that natural log has its derivative also in the integral. So I’m going to substitute for that. And the derivative is 1 over x dx. A perfect substitution. The w will take care of natural log, and the dw will take care of all the rest. But let’s take care of the limits of integration as well.
So when x is 1 what’s w? The natural log of 1 is 0. When x is e, we have a natural log of e. A natural log of its own base, that’s going to be 1. So the new limits are 0 and 1. This becomes w and this becomes dw. So the integral of w is ½w². So I’ll take this from 0 to 1. First at 1, ½ of 1² is ½ of 1, minus ½ of 0² which is just 0. The answer is just a half. That’s it.
So it’s really easy if you switch the limits over, to just go ahead and evaluate the integral with respect to w. And then plug in the limits.