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These are the user uploaded subtitles that are being translated: 1 00:00:00,360 --> 00:00:06,570 Get a room to talk about how to evaluate the integral of an even function to complete this problem will 2 00:00:06,570 --> 00:00:11,670 confirm their function is even and then simplify and evaluate the integral. 3 00:00:11,970 --> 00:00:17,760 In this particular problem we've been asked to evaluate the integral of x to the 6 plus one on the range 4 00:00:17,850 --> 00:00:19,930 negative to the two. 5 00:00:20,430 --> 00:00:25,740 And the first thing we should notice with this problem is that we have the definite integral range here 6 00:00:26,070 --> 00:00:33,660 negative to the positive too which means that we're being asked to evaluate a symmetric area about the 7 00:00:33,720 --> 00:00:45,040 y axis so we have here the x axis and the y axis and we want to evaluate from negative to here to positive 8 00:00:45,060 --> 00:00:45,640 2. 9 00:00:45,750 --> 00:00:52,130 So the same distance from negative to the y axis as it is from the y axis the positive 2. 10 00:00:52,170 --> 00:00:58,490 So that's always our first give away when it comes to simplifying this as an even function. 11 00:00:58,500 --> 00:01:03,990 The next thing that we want to look for is to test whether or not our function actually is even if our 12 00:01:03,990 --> 00:01:09,570 function is in fact even and we're being asked to evaluate the same distance to the left of the y axis 13 00:01:09,900 --> 00:01:16,430 as we are to the right of the y axis and we can take advantage of a specific kind of simplification 14 00:01:16,440 --> 00:01:20,790 when it comes to evaluating the definite integrals of even functions. 15 00:01:20,790 --> 00:01:23,550 So we want to test to see whether or not our function is even. 16 00:01:23,550 --> 00:01:29,280 So first of all let's call our function f of x so we just take everything that's inside our integral 17 00:01:29,280 --> 00:01:34,420 here and say F X Ziegel to x to the sixth plus one. 18 00:01:34,440 --> 00:01:40,800 Now in order to test whether or not the function is even we want to go ahead and plug in negative x 19 00:01:41,130 --> 00:01:43,320 for x into our function. 20 00:01:43,310 --> 00:01:50,010 So everywhere we see X and our function will plug in negative X will get negative x to the sixth plus 21 00:01:50,160 --> 00:01:50,930 one. 22 00:01:51,240 --> 00:01:58,350 And when we simplify this we'll see that we actually get X to the sixth plus one whenever we raise negative 23 00:01:58,380 --> 00:02:00,280 x to an even power. 24 00:02:00,300 --> 00:02:04,230 We're going to end up with a positive sign here and just x to the sixth. 25 00:02:04,230 --> 00:02:10,740 So what we see is that we actually end up with the exact same function as we did when we started with 26 00:02:10,770 --> 00:02:12,260 our original function. 27 00:02:12,270 --> 00:02:13,980 So because these two are the same. 28 00:02:13,980 --> 00:02:18,720 The original function and the function we got after plugging in negative X because those two are the 29 00:02:18,720 --> 00:02:19,530 same. 30 00:02:19,530 --> 00:02:22,800 That means that f x is even. 31 00:02:22,830 --> 00:02:27,150 So we can go ahead and say F X is an even function. 32 00:02:27,150 --> 00:02:33,390 So now that we've proven that f of BAC's is even we can go ahead and evaluate the integral and take 33 00:02:33,390 --> 00:02:36,050 advantage of the simplification that we were talking about. 34 00:02:36,270 --> 00:02:42,570 So because the function is even and because we're evaluating from negative to the positive to the same 35 00:02:42,570 --> 00:02:48,390 distance on the left of the y axis as on the right of the y axis instead of evaluating from negative 36 00:02:48,390 --> 00:02:57,880 to the positive to we can go ahead and evaluate from 0 to 2 and just multiply by 2. 37 00:02:58,050 --> 00:03:03,240 And the reason is because since we know that the function is even that means that we know that the function 38 00:03:03,240 --> 00:03:09,090 is symmetrical about the y axis and if it's symmetric about the y axis that means we're going to have 39 00:03:09,090 --> 00:03:14,280 the same area in this range as we are in this range. 40 00:03:14,280 --> 00:03:21,330 And so we can just take the area on this range here and multiply it by two and it's the same thing as 41 00:03:21,330 --> 00:03:25,230 taking the area from negative to the positive two. 42 00:03:25,260 --> 00:03:33,990 So we'll simplify the integral and call it 0 to 2 multiplied by 2 x to the sixth plus 1. 43 00:03:34,020 --> 00:03:36,630 Now we can just go ahead and integrate. 44 00:03:36,750 --> 00:03:40,390 So we'll get 1 7 using power rule here. 45 00:03:40,410 --> 00:03:48,830 1 7 x the seventh plus X will be our integral and we're going to evaluate that on the range 0 to two. 46 00:03:49,020 --> 00:03:55,230 Remember that with definite integrals we always want to plug in the top number first and then subtract 47 00:03:55,230 --> 00:03:57,590 whatever we get when we plug in the bottom number. 48 00:03:57,750 --> 00:03:59,790 So let's go ahead and plug two in. 49 00:03:59,790 --> 00:04:12,450 So we'll get one seven times to the seventh power plus two minus two times 1 7 times 0 to the seventh 50 00:04:12,540 --> 00:04:14,310 plus 0. 51 00:04:14,400 --> 00:04:18,810 And obviously we can see here this entire second term will go away because we're just going to have 52 00:04:18,810 --> 00:04:19,710 zero. 53 00:04:19,830 --> 00:04:29,380 And in the first term here we'll have 2 times one seven times two to the seventh which is 128 plus two. 54 00:04:29,970 --> 00:04:38,460 Simplifying further will get 2 times 128 over seven plus and we can find a common denominator here and 55 00:04:38,460 --> 00:04:41,500 called to 14 over 7. 56 00:04:41,970 --> 00:04:44,510 That gives us 2 times. 57 00:04:46,460 --> 00:04:54,950 Over 7 which of course just simplifies to to 80 for all over seven and that's it. 58 00:04:54,950 --> 00:05:00,590 That is the value of the integral of x 6 plus one from negative to positive to. 6558