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Numerade Educator

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Problem 15 Easy Difficulty

Evaluate the integral. $\int \frac{x-1}{x^{2}+2 x} d x$

Answer

$I=x(\ln x)^{2}-2\left(x \ln x-x+C_{1}\right)=x(\ln x)^{2}-2 x \ln x+2 x+C,$ where $C=-2 C_{1}$

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Video Transcript

So here we are given examples of certain integral and this is in the form of partial fraction decomposition. So you have to do that process. Yeah. Which is equivalent to a over X plus B over X plus two or A and D are constants. Mhm Okay, so let's multiply both sides by X and x plus two. So this would be x minus one equals A times X plus two plus B times x. So let's group terms together. So A plus B, X plus two, A plus B. So we can see by just coefficients, A plus B is equivalent to one to A plus B equals negative one. So negative A minus B equals negative one. So A is equivalent to negative choose. So it is equivalent to negative to be would be equivalent to three in this case. So now we can apply our integral directly. So a was negative negative two. So negative two over X. Which is just a constant plus three over X plus two T X. So this is equivalent to negative two natural log of X Plus three. Natural log of a. Yeah exposed to. So this would be the same thing as the natural log of X to the -2. Which would be X In this form plus three natural log of X plus two. Or we can write as the que perform plus our integration constant. And this gives our final answer