Text description - screencast for exercise 13

[The narrator reads out the onscreen text.]

NARRATOR: Exercise 13. Differentiate f of x equals x times log e of x minus x, and hence find the indefinite integral of log e of x. So the derivative of x times log e of x minus x is going to require the product rule to differentiate the first part of this expression. So the derivative of x is 1 and multiply it by log e of x plus x times the derivative of log e of x, which is one on x, and then minus the derivative of x, which is 1. Simplifying each part gives log e of x plus 1 minus 1, which clearly simplifies to log e of x.

So we see that the derivative of x times log e of x minus x is log e of x. And if the derivative of x times log e of x minus x is log e of x, then we can reverse that statement and say that the antiderivative, or the indefinite integral, of log e of x is x times log e of x minus x. And, of course, because it's an indefinite integral, we need to add the plus c.