### Video Transcript

Determine the indefinite integral
of negative two divided by seven π₯ with respect to π₯.

In this question, weβre asked to
evaluate the indefinite integral of a reciprocal function. And we can do this directly by
recalling one of our integral results. For any real constant π, the
indefinite integral of π over π₯ with respect to π₯ is equal to π times the
natural logarithm of the absolute value of π₯ plus the constant of integration
πΆ. We can just note that our value of
π will be negative two over seven; however, sometimes it can be difficult to see
this. So, instead, weβll just take the
constant factor of negative two over seven outside of our integral. This leaves us with negative two
over seven times the indefinite integral of the reciprocal function, one over π₯
with respect to π₯.

And we know the integral of the
reciprocal function is the natural logarithm of the absolute value of π₯. We can just recall this, or we can
set our value of π equal to one into our integral result. Using either method, weβve shown
the indefinite integral of negative two over seven π₯ with respect to π₯ is negative
two-sevenths times the natural logarithm of the absolute value of π₯ plus πΆ. And itβs worth noting we can always
check our answer by using differentiation. Remember, when weβre finding the
indefinite integral of a function, weβre finding its most general
antiderivative. This means when we differentiate
our function with respect to π₯, we should end up with our integrand.

So, letβs evaluate the derivative
of negative two-sevenths times the natural logarithm of the absolute value of π₯
plus πΆ with respect to π₯. We can start by recalling the
derivative of the natural logarithm of the absolute value of π₯ with respect to π₯
is one over π₯. So, when we differentiate the first
term with respect to π₯, we get negative two over seven multiplied by one over
π₯. The second term is a constant, so
its rate of change with respect to π₯ is zero. This just leaves us with negative
two-sevenths times one over π₯, which we can simplify is negative two over seven
π₯. This is our integrand, so this
confirms that our answer is correct. Therefore, the indefinite integral
of negative two over seven π₯ with respect to π₯ is negative two over seven times
the natural logarithm of the absolute value of π₯ plus πΆ.