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JH
Numerade Educator

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Problem 14 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \ln (1 + x^2)\ dx $

Answer

$\int \ln \left(1+x^{2}\right) d x=x \ln \left(1+x^{2}\right)-2 x+2 \tan ^{-1} x+C$

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Video Transcript

Let's use integration by parts for this integral here. Let's just take you to just be the intern grand. So then go ahead and use the chain rule to differentiate us. So you get one over one plus x Square, but then take the derivative of one plus x squared. So that's our do you Then we're left over with TV equals the X. So vehicles X now using integration by parts here, recalled the formula U V minus and enroll video. So we'LL go ahead and plug in our use. Andy's here, so we have X natural log one plus X squared, so lets you times v and then we have minus in a girl VDO. So here we have the and then do you So let me go in and pull out that, too. And then X squared over one plus x squared. Now you may try partial fractions here, but first we see that they have the same degree. So here we would need to do polynomial division and we can go ahead and rewrite this. After doing the long division, we get one minus one over one plus x cleared. Yeah, and that is our partial fraction to composition, So there's no need to find the baby in the seas. And, well, we know how to integrate both of these. If the second one. If this if you don't remember what this one is, you could do a trousseau at sequel stand data. So after writing this, we have X ellen one plus X square minus two in a girl, one minus one over one plus x squared DX. So now X Ellen one plus x square. Then from this we have a minus two X when we integrate And then after doing the tricks up here, you'LL get a plus two times ten inverse x and then we'LL add our constant of integration, see? And that's your final answer.