Video: Finding the Integration of a Reciprocal Function

Determine ∫ (βˆ’2/7π‘₯) dπ‘₯.

01:14

Video Transcript

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

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 π‘₯ dπ‘₯. We then quote the general result for the integral of one over π‘₯. It’s 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 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 π‘₯ plus capital 𝐢.

And notice here I’ve written capital 𝐢 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 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 𝐢 is zero. Multiplying, and we end up with our derivative to be negative two over seven π‘₯, as required.

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