### Video Transcript

Integrate the natural log of π₯ dπ₯
by parts using π’ equals the natural log of π₯ and dπ£ equals dπ₯.

Well, this is great. Weβve been told to use integration
by parts. And weβve also been told to let π’
be equal to the natural log of π₯ and dπ£ be equal to dπ₯. So we recall the formula for
integration by parts. The integral of π’ times dπ£ by dπ₯
is equal to π’π£ minus the integral of π£ times dπ’ by dπ₯. Now we can rewrite dπ£ equals dπ₯
slightly. And we can say that if dπ£ is equal
to dπ₯, then dπ£ by dπ₯ must be equal to one. Our job is going to be to find the
missing parts of the formula. dπ’ by dπ₯ is fairly straightforward. The derivative of the natural log
of π₯ is one over π₯. And if we integrate both sides of
the equation dπ£ equals dπ₯, we obtain π£ as being equal to π₯. Substituting everything into the
formula, and we get π₯ times the natural log of π₯ minus the integral of π₯ times
one over π₯ dπ₯.

Well, this integral simplifies
really nicely. Weβre actually going to be
integrating one with respect to π₯. The integral of one is, of course,
simply π₯. And of course, since this is an
indefinite integral, we must remember to add that constant of integration π. So we obtain the integral of the
natural log of π₯ dπ₯ as being π₯ times the natural log of π₯ minus π₯ plus π. And integration by parts was really
effective for integrating the natural log of π₯. Because the derivative of the
natural log of π₯ is much simpler than the original function the natural log of
π₯.