WEBVTT
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Let's start off here by doing the u sub.
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Let's take you to just be x square, then
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d u equals two x dx and we can solve
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for X here, so X equals square you.
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So to rule u d x and then go ahead
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and solve this equation for DX. You have d
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x equals d you over to radical you. Now
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let's rewrite this integral. So we have a one
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over and then first thing we see is the X
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Well, we already rewritten X over here. So
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let's just write this as you to the one half
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and then instantly have a radical so we can just
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rewrite. This is the square root of four.
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You plus one. Yeah. And finally DX.
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Yeah. Do you, over two years to the
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one half? Yeah. And let's simplify this.
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We could pull out this, too, and then
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combine those you to those radical use in the bottom
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to get one over you times for you. Plus
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one. Uh huh. So a different looking in
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our girl. Little simpler looking in a girl.
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Let's take another use up here this time V equals
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for you plus one in the radical. The D
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V is two over the radical. So this is
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to over v d u and then solve this for
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, do you? So we just solved this equation
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up here for do you? And then you get
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this V over to D V Also, do you
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see this YouTube out here outside the radical? So
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let's solve this for you. So we'll get U
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equals V squared minus one all over four. So
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we have one half integral one over, and then
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the u we just found that over here and then
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by definition of V, this is just the right
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here. So times V and then we have our
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d you at the very end. And that's just
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V over to D. V right now. Before
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we had a great let's simplify. We have a
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four here on the bottom, but this four will
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end up at the numerator. So those cancel and
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we see that these vis cancel. So we just
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have the integral of the square minus one. And
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the first thing you should do here is just factor
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that before you try the partial fractions, of course
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, you could also do a tricks up here.
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If you wanted to, that would work. So
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this is the alternative. So here, if you
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use partial fractions, you'll end up with the composition
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of the forum. A over B minus one B
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overview plus one. So you have to go ahead
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and find a and B. And when you do
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so you're getting one half or a and then minus
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one half for me. So I just factor out
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that mine is there and then integrate these natural algorithms
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. Don't forget the absolute value. And also don't
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forget the plus. See at the very end.
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Now we're almost finished. Let's go to the next
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page and just rewrite what we previously had so here
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. V minus one minus one half l n V
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Plus one plus e and then V by definition radical
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of for you plus one. So let's replace all
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the vis with the radical and then recall the definition
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of you are first substitution for the original problem.
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So go ahead and replace the use with X squared
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if you want. Here you can drop the absolute
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value because the expression inside is positive. But if
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you're unsure, just go ahead and leave that absolute
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value there. Oops. So with that, plus
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ones inside the radical here and then we have another
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plus one on the outside. Add that constant of
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integration, See? Yeah, and that will be
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our final answer.