Question Video: Finding the Integration of a Function Involving Logarithmic Function Using Integration by Substitution Mathematics • Higher Education

Video Transcript

Determine the indefinite integral of negative three over 𝑥 multiplied by the natural logarithm of eight 𝑥 with respect to 𝑥.

Now, although this may look like a tricky integral to evaluate, it is in fact in a form which we know how to integrate. If we let 𝑓 of 𝑥 be equal to the natural logarithm of eight 𝑥, then we can differentiate 𝑓 of 𝑥 using the fact that the differential of the natural logarithm of 𝑥 is one over 𝑥 in order to find that 𝑓 prime of 𝑥 is equal to one over eight 𝑥. And then, since this is a composite function, we have eight 𝑥 inside the function of the natural logarithm. We mustn’t forget to multiply by the differential of eight 𝑥, which is just eight. This is because of the chain rule. Simplifying, we can obtain that 𝑓 prime of 𝑥 is equal to one over 𝑥.

Now let’s rewrite our integral. If we multiply the numerator and denominator of our fraction by one over 𝑥, then we can rewrite our integral as the integral of negative three over 𝑥 over the natural logarithm of eight 𝑥 with respect to 𝑥. And now we can factor the negative three in the numerator. And once we’ve reached this stage, we notice that this is in a form which we know how to integrate. Since it’s of the form the integral of 𝑎 multiplied by 𝑓 prime of 𝑥 over 𝑓 of 𝑥 d𝑥. Where our 𝑓 of 𝑥 is the natural logarithm of eight 𝑥. And our 𝑓 prime of 𝑥 is one over 𝑥. Therefore, our value of 𝑎 is negative three. Now we know that this integral evaluates to 𝑎 multiplied by the natural logarithm of the absolute value of 𝑓 of 𝑥 plus 𝑐. And we can simply substitute in the values of 𝑎 and 𝑓 of 𝑥 to find our solution. Which is negative three multiplied by the natural logarithm of the absolute value of the natural logarithm of eight 𝑥 plus 𝑐.