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JH
Numerade Educator

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Problem 76 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{x^2}{\sqrt{x^2 + 1}}\ dx $

Answer

$\frac{1}{2} x \sqrt{1+x^{2}}-\frac{1}{2} \ln \left(\sqrt{1+x^{2}}+x\right)$

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Video Transcript

Let's start this one off by doing the tricks up X equals Santana. Then if this is our ex Lady X Seek and square Seita data, so are in a girl can be written while first we see there's a X squared that's ten square and then we see t X. But we were re evaluated that that's just c can't square. And then in the denominator we have X squared plus one in the radical, usually one of Europe with a green identities. We could write that a c can't square and they're seeking We could cross off that seeking in the denominator. And we could also rewrite tangent as seek and squared minus one. So we're left over with Tangent Square here so that seek and square minus one and then we still have one worse he cannot hear. So that's in a girl seeking Kim theta d theta minus in a girl seek and Dana Dana. Now this second and our girl we memorized by now this is just a trigger and her girl here. But for the first one, we could do an aggression by parts here, so I won't go over the whole details here But you could take you to B. C can't data and then take TV to be seeking square theater. And so, for the first in a girl, when we evaluate that for the sea can cute and then we have one half at lend. So this is all for be integral the sea cans in a third hour. That was what we wrote on the previous page. And then here we subtracted in a girls he can't, which is natural log of seek and eight of those Xanterra and was good and add that constancy. And so this term here was just the term on the previous page seek and data and that out there, she had a date or two. So here just combined thes two terms here and we're basically finished after we draw the triangle. Of course. So we thought a little work to do here. Okay, so now we draw a triangle. There's our angle data. And from the previous page we had tangent was defined as just X. You're getting sloppy here. We had X equals Tan Dana, which we could also write his ex over one so opposite, over adjacent and then used with agreeing to Europe to find the high part news. So here, But all we have to do that was fine. See Canon Tangent, where no tension is just X and then c can of data is just hype on news over adjacent. So it's just the radical. So let's just go ahead and replace. He can't intention on the next page. So they're seeking in Tan. And then here we have the one half and the natural log of C can't plus hand. So there's he can't right there. The radical that we just found on the previous page tangent was X. Let's add that concept of integration, see? And there's a