Find the integral of the natural log of 𝑥 d𝑥.
To solve this integral, we’re going to use integration by parts. The formula that we’ll use is the integral of 𝑢 times d𝑣 d𝑥 d𝑥 is equal to 𝑢 times 𝑣 minus the integral of 𝑣 d𝑢 d𝑥 d𝑥. So notice we only have the natural log of 𝑥. That can be 𝑢. And then we can actually take natural log of 𝑥 as natural log of 𝑥 times one. This way, one will represent d𝑣 d𝑥.
So we wanted to use a 𝑢 that we can take the derivative very easily and then take d𝑣 d𝑥 as something that we can take the entire derivative very easily. So taking the derivative of 𝑢 with respect to 𝑥, we get one over 𝑥, because the derivative of the natural log of 𝑥 is one over 𝑥. And then the antiderivative of one would be 𝑥. So 𝑣 would be equal to 𝑥.
So now let’s plug this into our formula. Notice we have 𝑢 times 𝑣, which would be the natural log of 𝑥 times 𝑥. However, written like this, you may be tempted to write natural log of 𝑥 squared, which is not true. So it’s safer to write it as 𝑥 times the natural log of 𝑥. And we have minus the integral of 𝑥 because that’s what 𝑣 is equal to. And then d𝑢 d𝑥 can be replaced with one over 𝑥 d𝑥.
With our integral, the 𝑥s cancel. And we have the integral of one. And the integral of one with respect to 𝑥 is 𝑥. Therefore, we have 𝑥 natural log of 𝑥 minus 𝑥. But don’t forget to add the plus 𝐶. Therefore, our final answer will be 𝑥 natural log of 𝑥 minus 𝑥 plus 𝐶.