Question Video: Finding the Integration of a Function Involving Using the Laws of Logarithms Mathematics • Third Year of Secondary School

Video Transcript

Determine the integral of negative nine times the natural logarithm of 𝑥 to the eighth power all divided by five times the natural algorithm of 𝑥 with respect to 𝑥.

In this question, we’re asked to evaluate the integral of the quotient of two logarithmic functions. And we don’t know how to evaluate this integral directly. This means we’re going to need to try and rewrite this in a form which we can integrate. We’ll try and do this by using our laws of logarithms.

The first thing we need to notice is we can rewrite our numerator by using the power rule for logarithms. Recall, for the natural logarithm, this tells us the natural logarithm of 𝑥 to the power of 𝑎 is equal to 𝑎 times the natural logarithm of 𝑥. In our case, the value of 𝑎 is equal to eight. So by applying the power rule for logarithms, we can rewrite our integrand as negative nine times eight times the natural logarithm of 𝑥 divided by five multiplied by the natural logarithm of 𝑥. And we can simplify this. First, nine times eight is equal to 72. This means we’ve rewritten our integral as the integral of negative 72 natural logarithm of 𝑥 divided by five natural logarithm of 𝑥.

Next, we want to cancel the shared factor of the natural logarithm of 𝑥 in our numerator and our dominator. And it’s worth pointing out to cancel these, we need to assume the natural logarithm of 𝑥 is not equal to zero. In other words, 𝑥 is not equal to one. So we need to keep this in mind for the values of 𝑥 our answer will be valid for.

So by canceling these, we get the integral of negative 72 over five with respect to 𝑥. And this is a constant, so we could evaluate this integral by using the power rule for integration. Or we could recall the derivative of negative 72𝑥 over five with respect to 𝑥 is equal to negative 72 over five, meaning this is an antiderivative of our integrand. And of course, we need to add our constant of integration 𝐶. And this is our final answer.

Therefore, by using our laws of logarithms and simplifying, we were able to show the integral of negative nine times the natural logarithm of 𝑥 to the eighth power divided by five times the natural logarithm of 𝑥 with respect to 𝑥 is equal to negative 72𝑥 over five plus our constant of integration 𝐶.