### Video Transcript

Determine the indefinite integral
of negative two over seven π₯ dπ₯.

Weβre going to begin by removing
the constant factor from this expression. And that gives us negative
two-sevenths times the integral of one over π₯ dπ₯. We then quote the general result
for the integral of one over π₯. Itβs the natural log of the
absolute value of π₯. And so, we obtain our integral to
be negative two-sevenths times the natural log of the absolute value of π₯ plus
π. Finally, we distribute our
parentheses. And we find that the solution to
this question is negative two-sevenths times the natural log of the absolute value
of π₯ plus capital πΆ.

And notice here Iβve written
capital πΆ to demonstrate that the original constant has been multiplied by negative
two-sevenths, thereby changing it. Itβs useful to remember that we can
actually check our solution by performing the reverse process, by
differentiating. The derivative of the natural log
of π₯ is, of course, one over π₯. So, the derivative of negative
two-sevenths times the natural log of π₯ is negative two-sevenths times one over
π₯. And the derivative of a constant πΆ
is zero. Multiplying, and we end up with our
derivative to be negative two over seven π₯, as required.