Question Video: Integrating the Natural Logarithm Function by Using Integration by Parts Mathematics

Video Transcript

Integrate the natural logarithm of 𝑥 with respect to 𝑥 by parts using 𝑢 is equal to the natural logarithm of 𝑥 and d𝑣 is equal to d𝑥.

In this question, we want to evaluate an indefinite integral by using integration by parts. We can do this by first recalling integration by parts tells us the indefinite integral of 𝑢 times d𝑣 by d𝑥 with respect to 𝑥 is equal to 𝑢 times 𝑣 minus the indefinite integral of 𝑣 multiplied by d𝑢 by d𝑥 with respect to 𝑥.

Usually when we’re using integration by parts, we first need to determine which function we set as 𝑢 and which one we set as d𝑣 by d𝑥. However, we’re already told this information in the question. We’re told to set 𝑢 equal to natural logarithm of 𝑥. And we’re told in terms of differentials, d𝑣 is equal to d𝑥. This means that d𝑣 by d𝑥 is equal to one. We can then see if we set 𝑢 equal to the natural logarithm of 𝑥 and d𝑣 by d𝑥 equal to one, then in our integration by parts formula we get the indefinite integral of the natural logarithm of 𝑥 with respect to 𝑥.

So to apply this, we’re going to need to find expressions for 𝑣 and d𝑢 by d𝑥. Let’s start by differentiating 𝑢 with respect to 𝑥. That’s the derivative of the natural logarithm with respect to 𝑥, which we know is the reciprocal function d𝑢 by d𝑥 is one over 𝑥. Since d𝑣 by d𝑥 is equal to one, the derivative of 𝑣 with respect to 𝑥 is equal to one. In other words, 𝑣 is an antiderivative of one. And we know the derivative of 𝑥 with respect to 𝑥 is one, so we’ll set 𝑣 equal to 𝑥.

Now, we just substitute all of these expressions into our integration by parts formula. We get the indefinite integral of the natural logarithm of 𝑥 with respect to 𝑥 is equal to 𝑥 times the natural logarithm of 𝑥 minus the indefinite integral of 𝑥 multiplied by one over 𝑥 with respect to 𝑥. And we can simplify this; 𝑥 multiplied by one over 𝑥 is just equal to one. So, we’re just left with 𝑥 times the natural logarithm of 𝑥 minus the indefinite integral of one with respect to 𝑥. And we can evaluate this integral. It’s equal to 𝑥 plus the constant of integration 𝐶.

Therefore, we were able to show the indefinite integral of the natural logarithm of 𝑥 with respect to 𝑥 is equal to 𝑥 times the natural logarithm of 𝑥 minus 𝑥 plus 𝐶. And integration by parts was really useful for helping us evaluate this integral because the derivative of the natural logarithm of 𝑥 is a much simpler expression than its integral.