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.
