### Video Transcript

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

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 π of π₯ be equal to the
natural logarithm of eight π₯, then we can differentiate π of π₯ using the fact
that the differential of the natural logarithm of π₯ is one over π₯ in order to find
that π prime of π₯ is equal to one over eight π₯. And then, since this is a composite
function, we have eight π₯ inside the function of the natural logarithm. We mustnβt forget to multiply by
the differential of eight π₯, which is just eight. This is because of the chain
rule. Simplifying, we can obtain that π
prime of π₯ is equal to one over π₯.

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