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.