# Video: Finding the Integration of a Reciprocal Function

Determine ∫ (−2/7𝑥) d𝑥.

01:14

### Video Transcript

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.