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Evaluate the integral.
$ \displaystyle \int \frac{ax}{x^2 - bx}\ dx $
$$\int \frac{a x}{x^{2}-b x}=a \ln |x-b|+c$$
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 4
Integration of Rational Functions by Partial Fractions
Integration Techniques
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
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let's evaluate the integral of X divided by X squared minus B X. We see that there's a quadratic in the bottom in that denominator, so always look to see if that could be factored here. Fortunately, we could take another X and then in this case, there's no need for the partial fraction to composition because we could just go ahead and cancel the exes. And we just have a over X minus B. So if you want, this is R and read, this is our partial fraction to composition. We just have a constant over a linear term. Explain his B So always factor when you can doesn't happen all the time, but it could make the problem easier. Now, at this point, we have an integral that you've seen before. If this X minus B is was throwing you off in the denominator, they'd just go ahead and take a use up. So let's go ahead and do this Use up and that are in rule becomes a over you and then D s. It's just do you and we know this to be eight times natural log absolute value of you. Don't forget the constancy of integration and then, at this point, just back substance. Who you for X minus B a natural log X minus B plus he and there's a final answer. Go.
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