Determine the indefinite integral
of negative two over seven 𝑥 d𝑥.
We’re going to begin by removing
the constant factor from this expression. And that gives us negative
two-sevenths times the integral of one over 𝑥 d𝑥. We then quote the general result
for the integral of one over 𝑥. It’s the natural log of the
absolute value of 𝑥. And so, we obtain our integral to
be negative two-sevenths times the natural log of the absolute value of 𝑥 plus
𝑐. 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 𝑥 plus capital 𝐶.
And notice here I’ve written
capital 𝐶 to demonstrate that the original constant has been multiplied by negative
two-sevenths, thereby changing it. It’s useful to remember that we can
actually check our solution by performing the reverse process, by
differentiating. The derivative of the natural log
of 𝑥 is, of course, one over 𝑥. So, the derivative of negative
two-sevenths times the natural log of 𝑥 is negative two-sevenths times one over
𝑥. And the derivative of a constant 𝐶
is zero. Multiplying, and we end up with our
derivative to be negative two over seven 𝑥, as required.