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AP Calc 4.7b: Integration of Inverse Trigonometric Functions

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hello everybody Welcome To Part B of chapter 4.7 we're going to be going over today I using completing the square as a strategy to rewrite integrals blessing to be able to understand that sometimes okay so let's jump right into exams an example of completing the square completing Square okay okay so for completing the square X review what that even means basically it just means that he has a quadratic like an x squared a ZX twin right here to things you can ask him we can do a squat parentheses over 2 squared over 2 square this right here is where we you made something interesting happen before we kind of threw this piece in there which is allowed simplified this that minus that would be we're not doing anything that isn't allowed in algebra what does allows us to juzo is rewrite the quadratic in this falling squared plus BX over squared over c squared plus c usually end up being a single Factor perfect square trinomial factoring out to beat would just be an x squared + 2 squared right as we were perfect square shrine this is what we can do with it when we still have whatever that ends up being for 2 squared plus X the savior's to find the integral of x squared minus 4x + 7 the square WhatsApp for taking for that denominator right this denominator we're going to complete the square with we're going to take that back squared minus 4X Class 7 right exactly as I get up here I'm going to start with my x squared minus 4X I'm going to add I'm going to do + be over to square I'm going to do 4/2 squared -4/2 squared + 7 + again just to kind of reiterate visually what's happening here this is what completing the square is I'm just putting plus and minus what happens if I simplify this squared - 4 + -4/2 is -2 -2 * 2 is positive for who's going to be positive for so4 + 7 is going to be a positive 3 once again is a perfect square trinomial which can be factored as an S and then I use this sign right here minus the square root of the last name a 2 squared + 3 can I substitute the x / x - 2 squared + 3 this is really close to the Arctic and girls are trying to say is radical 3 the square root of the same and that you would just beat a hoe - 2 information girl run over arctangent over x minus 2 over let's try another one French phrase a little bit differently but we're still doing the same process region bounded by digraph 1 / radical 3 x minus x squared graph the X equals 1 equals 9/4 things that are actually like an easier thing to identify for our inverse trig integrals again I want to the stepfather doing hair which is complete the square a squared plus furthermore I can take x squared minus 3 complete the square with this PC with our negative x squared minus 3x so here's what we can set that up as we can have a negative x squared minus 3x is when my value is going to come in handy virtue squared -3/2 squared if you want to visualize what the constant would be plus $0 you can see why that's redundant. I am going to get rid of that I bet that's what this whole thing is as the quadratic Square trinomial so I can have a negative and I can take the square of the first thing I asked I use this sign here and ask where it is it's in the square root of this really is just 3/2 * 3 / 2 call the pole Factor Square that's how it works out perfect square trinomials factored in this way 3/2 Square I did have a snack on the outside of everything so right here was taken care of when I just over whatever this is good enough for us for now we can see and we can substitute this into the denominator under the radical hour the radicals are function now we're going to have power 3 x squared minus -3 squared where are bound are from 1/2 to 9/4 so we say 1/2 to 9/4 because those are the X the line just say area equals for this what we have antiderivative of arcsin you over equals you just going to be for 3 or 2 3/2 / 3 has an indefinite the definite integral some going to have to use the fundamental I'm going to have to wait it are there was a typo in here at all you guys to fix it was actually supposed to be from the lines x equals 3 down here in the right are bound should be friends 3 have and I'm right here it should be as well okay and now what we do is just like we always handle the fundamental theorem of calculus we're going to our bounds and we're going to stuff them into function for X sir going to take 9/4 I'm going to say 9/4 - and right now I'm going to do something I'm going to change that to a fraction over forces gets a little bit easier make a call at 6 / 4 still over 3 hats this evaluated at the three have so far all over 3 hats text simplify here I can do 9 - 6 ft 3/4 divided by 3/2 so I'm doing 3/4 2/3 * 3/4 * 2/3 is going to be excuse me 6 / 12 and when I do 6 or 12 it's really just 1/2 -3 house I get zero and 070 and half and makes the entire thing zero and when I do those trigger valuations and I subtract I end up getting over 6 as my solution have just one more little quick side of examples to go through for you and then I'll leave you alone College top integration problems we had a red X over X radical x squared minus one this is South pretty easily she gets in Brawl for all that we are absolute value x + C thank you that's Grace and problems really similar but different squared our strategy should be restitution that will take care of it and we can call you squared minus one we can call a 2X Wichita how to put a 2in to are in a if we have to put a 1/2 on the outside with have one half over which is the same as 1/2 integral you furthermore when you use the power rule of integration 1/2 outside you to the positive 1/2 / 1/2 to the 1/2 / 2 and when I multiply on Uber for hold up is my issue should just beat you to the 1/2 over here and a 1/2 on the bottom to cross off to become one and I end up getting you to the 1/2 + C when I substitute x squared -1 to the 1 and that would be my antiderivative if we had it in a girl X over radical x squared minus take us to think about that do we have a rule that takes care of that the answer is no new rules would take care of that for us to be able to create try to do any other way because we have like this whole teeth if we had just radical x squared or just like radical Acts or something we could because we have radical not some can't take care of so we can just say nope can technically be done but not until much later chapters in calculus so that's it for today's video and I will see you guys in class tomorrow

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AP Calc 4.7b: Integration of Inverse Trigonometric Functions

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