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JH
Numerade Educator

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Problem 68 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{x^2}{x^6 + 3x^3 + 2}\ dx $

Answer

$\frac{1}{3} \ln \left|\frac{x^{3}+1}{x^{3}+2}\right|+C$

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Video Transcript

Let's start off here with the use of Let's take you to be x cubed, then do you? Over three is X square D X and that's exactly what we see in the numerator over here. X squared dx. So also, we have to rewrite this exit six is X cubed square so that you square so on the bottom. Let's do the denominator first. Excellent. Six three x cubed plus two in on the top where he saw that this is just do you and then we have to divide by the three Let's pull off the one third. So now this is a much better looking at a girl because weaken factor that denominator And then we could do partial fraction to composition here. So in this case, you want to find numbers and be such that we can rewrite the immigrant to be this, and it turns out that we have b who's won. So we have won over you, plus one and then we have a is minus one. All right, so use partial fraction the composition match up the coefficients to find your A and B as we did in the previous section. God and evaluate thes. That's a three out there, plus one minus. It's one thing I heard and then plus two. So here I just distributed the one third and generated both of these. I added that constancy of integration. So here we could do two steps. So pull out the winter and then I rewrite the longer of them as a fraction because its attraction and then I just replace you with execute. So on top of you, plus one so X cubed, plus one and then on the bottom X cubed, plus two. And then let's add in our constant of integration, see? And there's our final answer.