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JH
Numerade Educator

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Problem 47 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int x^3 (x - 1)^{-4}\ dx $

Answer

$$\ln |x-1|-3(x-1)^{-1}-\frac{3}{2}(x-1)^{-2}-\frac{1}{3}(x-1)^{-3}+C$$

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Video Transcript

Let's use a U substitution. Here, let's take you. It's just B X minus one motivated by this term here with the negative exponents. Then do you equals the X? So here we can come and rewrite. This of the first term is execute. So first off this rex, so x cubed becomes you plus one cube and then we have you to the minus four so we can rewrite. This is you plus one cute. You know the four and then the next step here we can just go ahead and expand that numerator. So go ahead and just cube this you plus one. And then the next step here would be to just split this up into foreigner girls. So for fractions here for the first fraction, we just have one over you Next. Next Fraction three, let's write. This is Yoo to the minus two. And then for the next one three year, the minus three. And for the last one, you to the minus four. So here we just use the power rule four times. First in the rule that's natural log absolute value, and then next will have negative one over negative three over you Excuse me, then minus three over to you squared. And then he's here for the last one. You'd have negative one over three you killed. So, sir, Constants of integration. See, the last step here is just to replace all of the EU's with X minus one. So we have will do. This is our last step. So that's the use Where? Right there and in minus one over three. And then you cube here, plus our constancy, and that's your final answer.