WEBVTT
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Let's try the substitution you equals Eat of X.
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The goal here is to use a substitution to rewrite
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the inner grand one over one plus needed the X
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as a rational function. So taking this to the
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U Do You is here The X T X,
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which I could write, is you DX and then
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divide by you. Take it, do you over
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you equals DX. So in the original integral,
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I can replace DX with do you over you and
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then in the denominator, we just have one plus
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you. So let's clean this up a little bit
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. This is just one over and then we have
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you one plus you to you. And that's a
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rational function that they asked for. So taking this
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rational function, let's go ahead and do partial fraction
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to composition, using what the author calls case one
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you distinctly of actors. Let's go and multiply both
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sides by that denominator on the left, and then
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let's go ahead and simplify by pulling out of you
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from the left hand side. The constant term is
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one on the right hand side. It's a so
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those are equal and because on the right we have
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a plus B in front of the U next to
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the U. But on the left, there is
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no you. So the coefficient in front of you
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, you must be zero. So solving this for
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me, we get B equals negative one and then
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we could plug These values in for Ambien are partial
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fraction and then we take this and this is the
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term that we're replacing the fraction with. And then
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we integrate. Let's go to the next place to
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do that. They was one so integral over you
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, minus one for B. So we pull out
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the minus one plus you, do you the first
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general natural log? Absolute value. Second rule,
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absolute value. One plus you. Now that one
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Plus he was bothering. You feel free here to
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do a use up. Let's say w equals you
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plus one, and that should help you evaluate the
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integral with that said the last step here. Replace
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you with how it's to find needed X. You
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could drop the absolute value here since you two,
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the exes always positive. And you could also drop
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it here because either the X plus one is positive
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, the last possible thing we can do here is
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to use the fact that Ellen X in need of
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X r inverse functions. So remember, if you
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have in Versace and you composed them together, you
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always get X. And that's exactly what we have
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here. Natural Log is being composed with either the
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ex, so that is just X. And then
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everything else is the same, and that's your final
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answer.