### Video Transcript

Find the integral of the natural log of π₯ dπ₯.

To solve this integral, weβre going to use integration by parts. The formula that weβll use is the integral of π’ times dπ£ dπ₯ dπ₯ is equal to π’ times π£ minus the integral of π£ dπ’ dπ₯ dπ₯. So notice we only have the natural log of π₯. That can be π’. And then we can actually take natural log of π₯ as natural log of π₯ times one. This way, one will represent dπ£ dπ₯.

So we wanted to use a π’ that we can take the derivative very easily and then take dπ£ dπ₯ as something that we can take the entire derivative very easily. So taking the derivative of π’ with respect to π₯, we get one over π₯, because the derivative of the natural log of π₯ is one over π₯. And then the antiderivative of one would be π₯. So π£ would be equal to π₯.

So now letβs plug this into our formula. Notice we have π’ times π£, which would be the natural log of π₯ times π₯. However, written like this, you may be tempted to write natural log of π₯ squared, which is not true. So itβs safer to write it as π₯ times the natural log of π₯. And we have minus the integral of π₯ because thatβs what π£ is equal to. And then dπ’ dπ₯ can be replaced with one over π₯ dπ₯.

With our integral, the π₯s cancel. And we have the integral of one. And the integral of one with respect to π₯ is π₯. Therefore, we have π₯ natural log of π₯ minus π₯. But donβt forget to add the plus πΆ. Therefore, our final answer will be π₯ natural log of π₯ minus π₯ plus πΆ.