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These are the user uploaded subtitles that are being translated: 1 00:00:00,360 --> 00:00:05,760 Today we're going to talk about how to evaluate the integral of an odd function to complete this problem 2 00:00:06,170 --> 00:00:11,110 will confirm that our function is on and then simplify and evaluate the integral. 3 00:00:11,520 --> 00:00:16,800 In this particular problem we've been asked to evaluate the integral of x squared sine X divided by 4 00:00:16,800 --> 00:00:23,280 the quantity 1 plus X and the sixth on the range negative PI over 2 to PI over to. 5 00:00:23,280 --> 00:00:28,180 So the first thing you want to notice about this integral is that you're being asked to evaluate the 6 00:00:28,180 --> 00:00:29,940 definite integral on the range. 7 00:00:29,970 --> 00:00:35,430 We'll call it negative a to positive A where A is a constant. 8 00:00:35,430 --> 00:00:42,580 So whenever you have the range negative a positive way where these two are equal to one another but 9 00:00:42,600 --> 00:00:46,690 the lower value here is the negative version of the upper value. 10 00:00:46,710 --> 00:00:52,000 You should consider checking to see whether or not your function is even or odd. 11 00:00:52,050 --> 00:00:57,360 The reason it's valuable to check to see whether the function is even or odd is because when you have 12 00:00:57,420 --> 00:01:04,200 the range here negative a positive A If the function is odd you know immediately that the value of your 13 00:01:04,200 --> 00:01:06,300 definite integral is zero. 14 00:01:06,480 --> 00:01:13,350 If the function is even you'll still be able to simplify it before you actually evaluate the integral. 15 00:01:13,350 --> 00:01:17,900 So because we have negative the positive A for our range here. 16 00:01:17,940 --> 00:01:22,690 Let's go ahead and check to see whether or not we have a function that's even or odd. 17 00:01:22,860 --> 00:01:32,250 So we'll call the function here f of x will say that f of x is equal to x squared sine X all divided 18 00:01:32,250 --> 00:01:36,390 by the quantity 1 plus X to the sixth. 19 00:01:36,390 --> 00:01:42,930 So given that that's our function we now want to evaluate to see whether or not this is an even or odd 20 00:01:42,930 --> 00:01:48,630 function way that we do that is by plugging negative x in for x. 21 00:01:48,840 --> 00:01:53,790 So plug negative x in for x everywhere we see X in our function. 22 00:01:53,820 --> 00:02:07,350 Let's go ahead and say negative x squared times sign of negative x all divided by 1 plus negative x 23 00:02:07,980 --> 00:02:08,940 to the 6. 24 00:02:08,940 --> 00:02:12,930 Remember that when you're looking to see whether a function is even or odd. 25 00:02:12,990 --> 00:02:19,620 You plug in negative x if the result you get back is f of x the same thing you started with then you 26 00:02:19,620 --> 00:02:25,820 know that your function is even if the result you get back is negative f of x. 27 00:02:25,830 --> 00:02:27,650 In other words the original function. 28 00:02:27,690 --> 00:02:32,110 But just with one negative sign out in front then you know that your function is odd. 29 00:02:32,130 --> 00:02:34,730 So we're going to be looking for either of those two things. 30 00:02:34,950 --> 00:02:42,090 In this case negative x squared just gives us positive x squared that negative sign cancels sign of 31 00:02:42,090 --> 00:02:48,180 negative x is actually the same thing as negative sign of X and we'll talk about why that's true in 32 00:02:48,180 --> 00:02:54,630 one second and then in the denominator here we have 1 plus negative x rays to the sixth power. 33 00:02:54,630 --> 00:02:59,220 Whenever we have negative x here raise to a to an even exponent. 34 00:02:59,370 --> 00:03:03,930 We know that we're going to end up with those negative signs cancelling and we'll just have a positive 35 00:03:04,320 --> 00:03:05,710 x to the sixth. 36 00:03:05,710 --> 00:03:12,660 So now when we simplify this we can pull the negative sign out in front the negative sign associated 37 00:03:12,660 --> 00:03:20,340 with the sign of X here we pull it out in front and we end up with negative x squared sine X all divided 38 00:03:20,340 --> 00:03:23,720 by 1 plus X to the 6. 39 00:03:23,880 --> 00:03:27,510 As you can see what we have here is f a vector. 40 00:03:27,540 --> 00:03:30,660 It's exactly the same as our original function. 41 00:03:30,660 --> 00:03:37,610 The only difference is that we have a negative sign here out in front which means we're looking at negative 42 00:03:38,100 --> 00:03:44,540 f of x and as we know here that means that our function is on. 43 00:03:44,710 --> 00:03:49,170 Now just a quick caveat here to return to the sign of negative x. 44 00:03:49,240 --> 00:03:55,120 Part of our problem if you're ever unsure whether or not you can simplify a Trigon a metric identity 45 00:03:55,120 --> 00:03:58,650 like this by pulling the negative sign out in front. 46 00:03:58,720 --> 00:04:03,400 You can always graph the function to see whether or not it's appropriate to call the negative sign out 47 00:04:03,400 --> 00:04:03,820 in front. 48 00:04:03,820 --> 00:04:11,980 So for example we know that the graph of sign of X if we look at an x y coordinate plane we know that 49 00:04:11,980 --> 00:04:21,260 the graph of sign of X if we have PI over to PI 3 PI over 2 etc.. 50 00:04:21,370 --> 00:04:26,470 The graph of sign of X looks roughly like this come down at PI. 51 00:04:26,470 --> 00:04:28,550 Here we have three PI over two. 52 00:04:28,660 --> 00:04:33,760 It comes up to me the x axis at 2 pi where this is 1. 53 00:04:33,800 --> 00:04:35,760 This is negative 1. 54 00:04:35,800 --> 00:04:42,100 So that's roughly the graph of sign of X to graph sign of negative x we can just take it one point at 55 00:04:42,100 --> 00:04:43,060 a time. 56 00:04:43,060 --> 00:04:50,320 So for example if we plug in zero for X to sign of negative x we get sign of negative zero. 57 00:04:50,320 --> 00:04:54,730 That's the same thing a sign of zero sign of zero we know is just 0. 58 00:04:54,730 --> 00:04:55,870 So we get that point. 59 00:04:56,290 --> 00:05:02,800 If we plug in PI over to two sign of negative x we'll get sign of negative PI over two. 60 00:05:02,830 --> 00:05:06,580 We know that sign of negative pyo or two is negative 1. 61 00:05:06,760 --> 00:05:09,190 So we get a value here of negative 1. 62 00:05:09,280 --> 00:05:17,350 If we plug in high to sign of negative x we'll get a sign of negative Pi which we know is 0 and we can 63 00:05:17,350 --> 00:05:18,360 continue doing that. 64 00:05:18,370 --> 00:05:25,160 And what we would see is that the graph of sign of negative x is this graph right here. 65 00:05:25,420 --> 00:05:33,460 And what this tells us is that we have exactly the opposite here so whenever the graph of sign of x 66 00:05:33,910 --> 00:05:39,580 is positive at this point let's say it's you know positive three fourths it's negative three fourths 67 00:05:39,580 --> 00:05:43,200 here when we're at positive one where it negative 1. 68 00:05:43,390 --> 00:05:46,760 So it's the same value except with sign of x. 69 00:05:46,780 --> 00:05:49,450 It's the positive value with sign of negative x. 70 00:05:49,450 --> 00:05:55,180 It's the negative value and that's how we know that it's OK to pull that negative sign out in front 71 00:05:55,180 --> 00:05:58,000 of the sign of x so you can always test it that way. 72 00:05:58,000 --> 00:06:04,660 But given now that our function is odd what we can deduce about the integral is that this whole integral 73 00:06:04,660 --> 00:06:07,330 here is equal to zero. 74 00:06:07,360 --> 00:06:11,380 Just by understanding that the function we had inside here was odd. 75 00:06:11,500 --> 00:06:13,490 We know that the integral is equal to zero. 76 00:06:13,510 --> 00:06:19,390 And the reason is because an odd function is symmetrical about the origin. 77 00:06:19,390 --> 00:06:24,400 So if you imagine something like this then I don't know what the graph exactly of this function looks 78 00:06:24,400 --> 00:06:24,660 like. 79 00:06:24,670 --> 00:06:30,970 But let's take a graph that symmetrical about the origin and maybe it looks like this. 80 00:06:31,030 --> 00:06:31,970 Right. 81 00:06:32,080 --> 00:06:39,070 If it's symmetrical about the origin as long as you're evaluating on the range Let's call this negative 82 00:06:39,120 --> 00:06:40,020 high over 2. 83 00:06:40,020 --> 00:06:47,980 So say as long as you are evaluating on the range negative PI over to 2 positive high over 2 you know 84 00:06:47,980 --> 00:06:55,670 that you're going to have the same negative area here to cancel out the positive area over here. 85 00:06:55,840 --> 00:07:01,550 And so because these two are going to cancel one another you know immediately that the area from the 86 00:07:01,630 --> 00:07:09,010 on this range is equal to zero because it's the same distance from this point to the y axis as it is 87 00:07:09,010 --> 00:07:15,580 from the y axis to this point the area to the left of the y axis has to be the same. 88 00:07:15,580 --> 00:07:20,410 But the negative version of this area on the right side of the y axis and they're going to cancel each 89 00:07:20,410 --> 00:07:20,940 other. 90 00:07:21,070 --> 00:07:24,180 Your integral will always be equal to zero. 91 00:07:24,250 --> 00:07:30,340 So that's it because we're evaluating here on the range negative a positive a. 92 00:07:30,460 --> 00:07:35,320 And because we've identified a function as odd the fact that we have all of those things means that 93 00:07:35,320 --> 00:07:38,610 we know that the integral here is equal to zero. 10453