WEBVTT
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Okay, let's go ahead and solve this integral right
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here. It's actually one of the very interesting cases
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off into goals. Um, we do this thing
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called a where Stress substitution. He's a very famous
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mathematician, and when you talk about one of the
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really rigorous field off mathematics called analysis, you'll be
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talking about him a lot. And he's very famous
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. Anyhow, He figured out that no matter which
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kind of tribunal metric expression you have when you use
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this, T substitution with tangent of X over to
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the expression will always end up looking like a rational
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function and rational functions. We know a lot of
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techniques to solved the integral, so hopefully we can
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make the problem look a little bit easier is basically
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the idea of this thing now from the previous problems
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, you must have already figured out or proved what
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cosine or the X looks like with this substitution,
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you know, co sign of X equals to one
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minus t squared over one plus the's word, while
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the X is to over one plus t squared DT
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. So according to this substitution, we can simplify
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this expression with a little bit of algebra that This
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is the integration off one plus T squared over two
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teeth. This is the 1/1 minus cosine X portion
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, and the D X is, of course,
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to over one plus t's word. And as you
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can see, the twos cancel each other out.
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The one plus T's court cancels each other out,
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so this is simply equal to, um, one
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over t squared the integration off that is negative on
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over t, of course. And then we're going
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to substitute back in what T was equal to.
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So what's the reciprocal of tangent? It's of course
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, co tangent. So you will have negative co
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tangent off X over to Well, let's not forget
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the fact that this is a an indefinite, integral
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, so constant should always come there. And this
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is how you solve this integral using wire stress institution
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.