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Evaluate the integral by completing the square and using Formula 6.

$ \displaystyle \int \frac{dx}{x^2 - 2x} $

$$\frac{1}{2} \ln \left|\frac{x-2}{x}\right|+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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Let's evaluate the integral by completing the square and using Formula Six. So I've written Formula Six here on the right from yourself. Book. Now it's completely square civil. Add one and then subtract one. We have X minus one Swear minus one. If we want with. Let's also right. This's once were due to the fact that we have a sweat over here on the right. So now we could rewrite this in a girl DX X minus, one squared, minus one square. And if that minus one is throwing you off because it's not showing up over here and it's up right, I could go ahead and do a use up here. U equals X minus one. Do it was the ex. Then this in rule, becomes do you use square minus one's where and now we use the formula here a equals one. So then we have one over two times one and then there We should not have an interval any more. On the right hand side, we're using Formula Six over here, So one over two times one natural log and then we have X minus a and then X plus a. So here we have you minus four. You plus one. And then we add Si. And then now we just go ahead, replace you with excusing this. So we have one half, and then we have X minus two over X plus our constancy, and that will be our final answer.

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