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How To Integrate Inverse trigonometric functions by completing the Square
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Feb 19, 2025
In today’s lesson , we’ll discuss how to find the integral of more complex trigonometric inverse functions by using the completing the square method
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so we're going to learn how to complete
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the square to solve trigonometric uh
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inverse functions right so if you did
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Algebra 2 you know how to complete the
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square so we're just going to pull this
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one out for now let's just pull it out
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so we have X2 -
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2x that's two you can set it up to zero
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if you want it's up to you whatever you
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want to do you don't have to right so
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what you do is you take this number in
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the middle and you divide it by two
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right so we going to take two out2
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by two that's -1 right we're going to
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square it that's step two so step one
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you divide it by two step one step two
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you square it right and that gives us
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one and then step three you add one and
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subtract one on both Sid so if you you
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going to go x² - 2x + 1 - 1 + 2 okay you
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don't have to take setal you don't have
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to okay so now what we have is we know
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what this is right if you were to fo if
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you were to find that that's x -1 2
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right so that just be x -1 2 and this is
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why2 +
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1+ one right so we go back here we
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replace it that give you 0 2 1 / X - 1
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2 + 1 s pretty much DX right and we know
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that this matches up well that over
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there a squ let me rewrite it so that
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way it looks better so this is pretty
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much 1 / 1 2 + x - 1 2 02 DX right so
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this is a square this is U sare because
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we know that du if you let X a is
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constant a is constant right so if you
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let u = x - 1 du is 1 DX so we don't
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even have to do anything here so
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basically we're just going to go ahead
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and apply the formula right so that's
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going to be 1 over U right which is 1 x
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-1 is it 1 a or 1 U I got to make sure
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I'm looking at the formula
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correctly it's one/
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a so it's one over one which is just one
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we don't even need to be just put it for
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the sake of people that are watching
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this R 10 right x -1 okay over one now
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it's not plus C here because we going
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between two and zero right so it's not
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going to be plus it's going to be two
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and we have that so it's pretty much R
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10 uh 2 - 1 which is one right minus
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Octan Z now you can put this in your
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calculator and find what this is and I
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should do the the job how would I do the
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AR work ar your calculator is this uh
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101 1
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yeah it's the inverse function the r 10
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is an inverse function of
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[Music]
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that
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so there's another problem that I like I
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like I want to work on you good on this
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one yeah all right so let's raise this
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real
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quick and see how we going to solve the
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next one it says 2x -
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3
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over square root of
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uh 4x - x² DX right so this is what we
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have here so how we're going to solve
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this one so yes there's several possible
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ways you can solve this but number one
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step is we always we going to try
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and complete the square here so let's
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just pull this out for now we have x^2 +
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4x right so what I'm going to do is I'm
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going to put negative 1 aside for now
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like this and I'm going to have X2 - 4x
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I need something here right so again I'm
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going to take this one divided by
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two right so -4 ided by two that's what
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that's -2 right -2 2 is
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4 right so I'm going to have your 4 - 4
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like that
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okay and now we can take this one out
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okay we going to take it out out of the
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why can't you just just let it let U be
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X let U be X this yeah U be like 4x - x
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s like a yeah a no it's not a s no it's
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not so it's not going to work right so
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then we have that and then we now we can
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go uh negative this is X uh - 2 s right
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and then because this a negative and a
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negative that would
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be+ 4 because this negative is going to
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change into a positive right now we have
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we can go back here and write 2x - 3
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over < TK of 4 right - x - 2 2 now we
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have something going on here we can kind
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of right so now the the rest is just to
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figure out how to transform this one
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here which we can do right we can let U
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equals to uh let's see if I let U equals
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to if I actually I can split this I can
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split this into 2x right over or just
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put the two outside if I want X over 4 -
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x - 2 2 right and - 3 and then I have uh
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just 1 over 4 - x - 2 2 DX now here we
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can use a u substitution
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here what are you going to do with
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X well if I you yeah if i u = x - 2
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right X will be What U +
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2 remember what we did in the first
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chapter we can use that here all right
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and then the rest is just you applying
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for this one you can apply the
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trigonometric uh one of those arct for
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this one okay and then that we just
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going to do that and then and solve it
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that should do the trick all right we
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good on that all right
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