WEBVTT
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Let's use integration by parts toe Prove the following formula
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and let's go ahead and denote the integral on the
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left. But I am so we see the end
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shows up on the right. Is the exponents in
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the denominator? So for our integration, my parts
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let's make this choice for you then do you?
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Is negative to an ex that you think the chain
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rule from calculus TV is the ex. So then
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, by using immigration my parts we know the formula
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. You ve todo So in our case, lets
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go out and write that out. Look, no
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, cancel those double negatives. Pull off the two
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one and that here are always the following trick.
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I'LL rewrite this as x squared plus a squared minus
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a squared And then I'll split it this up into
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two in a girls. So this is then we
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have X squared plus a square. Now we see
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that we could cancel one of these terms Run And
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then for the last in a girl Let's pull out
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that a square. We still have that too in
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from here. Good. And then in a girl
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one over x squared, plus a square. Now
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notice that we can rewrite these remaining two in a
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girls in terms of this interval right here. So
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this integral one over X square, plus a square
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to the end, is just I am. And
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here would you have him and plus one that will
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be I and plus one. So let's go to
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the next patient. Write this out in the first
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term here. This is from our UV two and
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to end I am minus two and a squared I
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and plus one that's good and soft Fur I n
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plus one the smiles over here. So we have
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this one minus to end. Yeah, no.
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And what's good and simplify this. Cancel all those
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double those minus signs and then here. Okay,
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so now there's new formula. This hole's for any
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end. So instead of using the n plus one
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on the left, let's just replacing within. So
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here, just since this works for in the end
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, let's replace and with and minus one that here
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is the left hand side just becomes I am the
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right hand side. Just ride it out. So
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this is the numerator and then on the denominator to
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a square. But this time and minus one going
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on to the next page. Simplify each term here
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, that's our first term. And then our second
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term Teo So remember, on the left this's all
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equal toe I end And then here on the left
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, I could just rewrite that first her. That's
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the term that we wanted from the formula. We're
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basically done here. The last step is it.
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Just replace Pull off this constant here and then replace
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i n minus one with the definition as thie Integral
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. So that's a DX X squared plus a squared
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all to thee and minus one power. And that's
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the formula that we attended on proving Now let's go
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ahead and start on part B. So let me
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go to the next page here. So we'LL plug
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in for actually let me just write out two Step
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the integral here for part B one over X squared
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plus one square DX. So here we see that
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take was one and then we have an equals two
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. So let's fight and use this in our formula
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. So we get X over to X squared plus
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one and then going to the second term here we
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plug in and equals two again. So we get
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one half and then in a girl, one over
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X squared plus one. Now, if the Senate
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rules bother you feel free to do it. You
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some here, our strengths up. Excuse me?
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No. And we could simplify this one half and
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then we have our Tana Vicks plus our constancy.
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Now there's two intervals in part B. This is
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our answer for the first animal. And we could
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actually use this answer from the next interval. So
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we're still in part be here. But the next
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integral this time we have X squared plus one.
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Cute. So using the formula from part eh?
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This time and equals three equals one. So we
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have X over We'LL have two times two x square
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plus once where and then the second fraction two times
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one times two one over one plus x clear square
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My notice this integral That just popped out. That's
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the interval we just evaluated on the last stage.
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So here was just plug in our answer from the
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other the previous part of part B And then we
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have three over four and then we can replace this
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by previous age X to X squared plus one and
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then we also had one half. It's an inverse
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X and let's got and add that constant of integration
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, see, and very in. And that would
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be our answer for the second in a girl party
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and that used the earlier part of the same problem
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. B. But it also used the formula from
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party, and that's our final answer.