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AP Calc 4.5b: Integration by Substitution

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Definite integrals

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you're welcome again we're going to talk about function you get back right we're going to find it High Acres continue on close interval and Fantasia rain Man say that the integral of x can be printed activity you and you equal the integral of a rate you evaluated DMV off of you they were given the power to evaluate and we are given the integral from 0 to 1 of act outside you can come here first of all be able to talk about yesterday but d'you is going to have to be to act black I also want you to say something over here that you what does Ulta equal to DMX where you at you don't mind using this as music create new limit turn down for integral 0 1 0 + 1250 equal 1/4 + 1 I'm not going to say G of one to be open me yeah squared plus 101 and a hare forget me we have our new Bonsai Garden right here we have you and we had a few I remember from yesterday is a pony and a girl we're going to say the in a girl from one the two right now and now we're going to say 2 or 1/2 time to act right now we have a deal add times x squared + 1 everything out we already put in our new room and now I'm the bu we had a square + 1 as are you and then we have that integral from 1 to 2 with the 1/2 out front I YouTube you can find the antiderivative outside the antiderivative famous saying 1/2 outside 10 / 4 - 1 / 4 and when I finish that I get 1/2 outside for - 1/4 so I get one outside 16/4 + 5 + 16 + 4 - 1 get me 15/8 and that is another one again our task is to evaluate the definite integral the phone is a little bit more on top I want you to recognize power here that's what we're going to make our you 2 x -1 now I want you to understand something in here the derivative of you right now would be too but there's really no way that I can make the actress I can't do anything so I can replace this asking you is to run are you a ride and now but someday if I went through something minus 1 and I have to add one plus one equals two another word find squared plus 1 over to save that for later with my you up there the act now is also going to be I want half outside square root of 1 once I have one outside I deactivate that would go away and then right here do you deactivate variables don't matter how to take care of the outside I would get it to you and then I would get a time that the inside call that be you what fun things can happen here are going to make each other go away 1 equals u d u d x I want to talk to deactivate so I can replace it with you and then I get that deck equals you help so far I have everything in place so I can start to substitute back in however I need to find the value waiting I like that by noon are going to come from at 1 when I throw a one in here and get radical 2 * 1 - 1 square root of 1 GM 502 * 5 - 1 radical 9 and that becomes three I have my integral is now going to be from 1 and I'm going to replace the speaker very carefully and I'm going to write it in terms of you as I solve 2 + 1 / 2 right now radical 2 X - 1 how many times do you have to do I have the same as you do alright I'm going to be organized but we're going to work with it again are we have one two three all over you and that is prime Eminem left with the kind of an easier in a girl 1/3 of you squared + 1 / 2 and furthermore can I get one half of a girl from 1 to 3 squared + 1 d u when I start taking antiderivative here I get 1/2 YouTube / 3 + you and valuing that from one screen and tell what I end up getting 3 + 3 2729 1/2 bracket 9 + 3 minor if I put A1 in here at one third work here 1/2 outside 12 - 4/3 further we get one half outside 32 River Street I am when I X my denominators I get 32 / 6 when I reduce the fraction I get 16 / Street let's try another one but that's something even more interesting name of the game to evaluate the definite we are getting a definite integral from 0 to 1 x squared outside radical one- and now I what I wanted to do here it's actually a radical key radical one- and I want you to remember God Bless America no problem in order to replace the  right here practice problems for us we get you squared equals 1 I could subtract 1 and then I could change the sign of everything once again, save that for later that's going to be autographed keep going with it right now we have something we can we have something we can substitute in for this radical over here Square D dx11 the derivative of negative here I sit outside in front and I'd x. -1 to get Negative to you and I have to worry about the inside the derivative of just you don't not here that part would go away again Axel x x DX and now I get back to you what are the three things I can use to back in I need some new women so I'm going to write those we're going to use our integration are going to come from that Deanna Albany to evaluate at zero that was our first found I get ready to go 1-0 going to be I meant why do GM one I get radical 1-1 10 I want you to understand that they're going to look a little bit different we write our new integral this one came first one and then we had zero that's okay we shouldn't at zero we got one we evaluated at 1 we got here at 4 that's exactly what the rest of our integral by our place Freddy Square I'm going to square stuff right here so I'm going to say negative squared + 1 squared article 199 call you and then black actually replacing with you negative to you alright we can start to simplify some things here and 1 negative squared + 1 squared times 2 times negative times a negative I wish I can turn into negative 2 squared do you still have negative you squared + 1 squared binomial here by square the first thing so I get you to the 4th and then to Times Square the last thing + 1 binomial expansion by -2 squared and bracket is what I'm dealing with at the end I want tickets to everything in here you know what would be here how to use squared 4 - 8 + 1 another step I could take care of everything in the trinomial 6 Manitou YouTube you squared when I create my own I have you to the 7th / 7 - 2 YouTube calling the power rule antiderivative and I'm evaluating from 1 to 0 so remember when were you going to fundamental theorem of calculus do that right now -2 Stone outside -1/7 - 2/5 + 1/3 bearded negative negative to everything outside of a -1/75 and a negative 1/3 common denominator inside to outside -8 over 105 - 2 x - 8 here we get me positive 16 over 105 and that is fully evaluated 1 month puppy I want to go or not even and I on integration in a non function in short if your stamp an even function then we can play the integral from a negative value 8 is equal to double that in 02 value big of a double positive half of an integral odd say the integral from negative to positive of that function it just and I want you to call what because I don't know it's been a while even happen how to do the negative X where function we should get a positive regular after that is if we get a negative Hoopa save your eye to evaluate and it went from negative pi over 2 and sine of x cosine X well I sign at consigner we're going to answer the question isn't even or odd we can do that by second shooting in what happened sign poop 9 pack negative minus sign I want you to see the difference between once we substituted in negative and our original function remember when Batman you're kind of right that I had with me according to our room here from negative to positive that same thing 730 integral from -pi/2 to pi/2 cosine x + sinx 14 x 0 open Quickoffice widest negative one and one we're going to call this negative pi over 4 here we would have a negative pi over 2 I'm right here you would have president number for baby going from -pi/2 down to my money would go back up after the origin they would go up to 1 at 5 or 4 and then across again at pi over 2 can I curve we got going on the area that happening over on the right side we have positive area and over in the left side we have negative area to area negative area on for the kind of cancel each other out perfectly and it will happen Okay I have to show you a few

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AP Calc 4.5b: Integration by Substitution

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