WEBVTT
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So today we will be solving the integral from 2
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to 6 of either the two x plus three y
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d X. Now the first thing we do when
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we look at this is we we can figure out
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howto integrate e to the X. Something simpler.
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So we want to use what's called u substitution.
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And to do that, we have to pick what's
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Are you gonna be So are you in this case
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is going to equal to X plus three. Why
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? And that means that our d'you has to equal
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, um, two times DX because three wise like
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a constant and so the derivative of that in terms
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of X will be zero. So now I have
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my you and might do so I can rewrite my
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integral here, and I'm going to write just a
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definite interval and then we'll bring back the two in
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the six. So this will be the mineral of
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E to the u times D'You And don't forget about
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this two over here, we have to make ah
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, 1/2 and I'm gonna bring this in the front
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. So since it's a constant, we can take
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it out, and it could be 1/2 times this
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interval of E to the U D U. So
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now we know how to integrate this because the inter
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role of either the U is just eat of the
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U. Um, if you don't remember this,
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it's also the interval of or it's e to the
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you and then over natural log of your the base
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. But the natural log of ease Just one.
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And so that's why he is a special function where
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when we integrate this, we just get 1/2 e
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to the U. Normally, if we were just
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solving the definite and a role we would put plus
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C at the end. But we don't need that
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here because we're gonna end up plugging in this,
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too. In this six sow, the seed would
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cancel. So I'm gonna write 1/2 e to the
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you, which I'm gonna replace my you back with
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two X plus three. Why? So say two
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X plus three. Why? And I'm gonna solve
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this from 2 to 6 or integrate this rate from
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from 2 to 6. So first, what we're
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gonna do is plug in the six for X.
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Okay, so we're gonna get 1/2 times E to
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the to 10 6 which is 12 plus three.
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Why and then minus. And now I plug in
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the two. So I have 1/2 again times E
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to the two times two, which is four plus
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three wives. And now we can write this a
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little bit nicer just by writing the 1/2 once.
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And you could have done this to start. But
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now we'll have 1/2 and then we'll do e to
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the 12 plus three. Why minus e to the
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four plus 31 And this will be our answer for
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our indefinite integral above right here.