Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:00,630 --> 00:00:03,700 Today we're going to be talking about how to evaluate an indefinite integral. 2 00:00:03,960 --> 00:00:10,950 And in this particular problem we've been given the integral of 3 x squared plus 2 x plus one dx and 3 00:00:10,950 --> 00:00:16,230 this is the most basic type of indefinite integral that we can deal with only because we'll only really 4 00:00:16,230 --> 00:00:22,700 need to reverse the power rule process that we learn from derivatives in order to evaluate it. 5 00:00:22,710 --> 00:00:27,900 You remember that when we were dealing with derivatives and we were using the power rule to find derivatives. 6 00:00:27,900 --> 00:00:31,240 What we have is something like this or we can even take this example here. 7 00:00:31,350 --> 00:00:36,720 If we had 3 x squared and we wanted to take its derivative. 8 00:00:36,720 --> 00:00:42,540 Remember that power rule told us that we would bring the exponent down here in front and we'd multiply 9 00:00:42,540 --> 00:00:44,660 it by whatever coefficient was already there. 10 00:00:44,850 --> 00:00:48,260 So in other words we'd get three times two. 11 00:00:48,360 --> 00:00:51,530 We'd leave the X there and then then we'd subtract 1 from the exponent. 12 00:00:51,540 --> 00:00:53,620 We'd end up with two minus one. 13 00:00:53,670 --> 00:00:58,650 And of course the result of that was a derivative of 6 x. 14 00:00:58,650 --> 00:01:03,350 When we're dealing with an indefinite integral that's just a polynomial like this. 15 00:01:03,480 --> 00:01:09,000 We're reversing the power rule process so this time instead of kind of dealing with the coefficient 16 00:01:09,000 --> 00:01:12,980 first and then the exponent second like we did with the derivative. 17 00:01:13,050 --> 00:01:17,310 This time we're going to be dealing with the exponent first then the coefficient and what I mean by 18 00:01:17,310 --> 00:01:22,220 that is instead of subtracting 1 from the exponent we're going to be adding 1 to the exponent. 19 00:01:22,230 --> 00:01:23,880 So the X is going to stay here. 20 00:01:24,120 --> 00:01:27,940 And instead of two minus one we're going to say two plus one. 21 00:01:28,190 --> 00:01:30,740 Then we take our new exponent in this case. 22 00:01:30,750 --> 00:01:35,820 Two plus 1 and we divide our existing coefficient by this new exponent. 23 00:01:36,000 --> 00:01:41,040 So our existing coefficient is three we leave that there but we divide by the new exponent which is 24 00:01:41,040 --> 00:01:42,340 2 plus 1. 25 00:01:42,360 --> 00:01:49,480 So this is going to be our integral and the result of this of course here we'll get 3 divided by three. 26 00:01:49,470 --> 00:01:50,730 So that's just going to be one. 27 00:01:50,730 --> 00:01:52,500 And this will disappear here. 28 00:01:52,770 --> 00:01:56,500 And all we're left with is now x to the third two plus one is three. 29 00:01:56,550 --> 00:02:02,310 We're left with X to the third as the integral of 3 x squared and that's how we're going to deal with 30 00:02:02,310 --> 00:02:04,160 each of these polynomial terms. 31 00:02:04,170 --> 00:02:06,430 So let's take a look at this integral here. 32 00:02:06,480 --> 00:02:09,890 We already know that the integral of 3 x squared is x cubed. 33 00:02:09,960 --> 00:02:14,340 And remember we're just taking this term by term one term at a time when we have a polynomial. 34 00:02:14,400 --> 00:02:17,010 We only need to deal with one term at a time. 35 00:02:17,010 --> 00:02:24,450 Now we move on to 2 x and again here for 2x what we're going to do is leave the existing coefficient 36 00:02:24,480 --> 00:02:26,820 to we've got the X here. 37 00:02:26,940 --> 00:02:28,920 This is X to the first power. 38 00:02:28,920 --> 00:02:33,840 So we're going to add 1 to the exponent we're gonna get one plus one and then we're going to divide 39 00:02:33,870 --> 00:02:38,400 our existing coefficient by the new exponent and put one plus one there. 40 00:02:38,400 --> 00:02:40,880 Now this is the technical slow way of doing it. 41 00:02:40,890 --> 00:02:46,650 As you practice you'll get much faster at doing this you won't have to add 1 to the exponent like this. 42 00:02:46,650 --> 00:02:51,750 Write it out and then divide by this expanded exponent year you'll get faster at it. 43 00:02:51,750 --> 00:02:54,450 But I just wanted to show you guys how it worked. 44 00:02:54,450 --> 00:03:00,570 Now when you get to the integral of a constant like this we have one here you can just know that the 45 00:03:00,570 --> 00:03:03,690 integral of one is X in a similar way. 46 00:03:03,690 --> 00:03:13,270 The integral of 3 is 3 x so the integral of 3 would be 3 x the integral of 1 would be 1 x or just x. 47 00:03:13,290 --> 00:03:19,400 If you had the integral of pi which is also a constant the integral would be pi x you get the idea. 48 00:03:19,620 --> 00:03:23,560 But the way to remember this it's the same reversal of power rule. 49 00:03:23,610 --> 00:03:24,920 This isn't one. 50 00:03:25,110 --> 00:03:27,890 It's one times x to the 0. 51 00:03:27,900 --> 00:03:32,860 Remember that X to the 0 is equal to 1 because anything raised to the power of 0 is 1. 52 00:03:32,880 --> 00:03:35,970 So this is really 1 times x to the zero power. 53 00:03:35,970 --> 00:03:41,040 And essentially what we're just doing here is the same process we've done here before we say plus the 54 00:03:41,040 --> 00:03:42,760 existing coefficient 1. 55 00:03:42,990 --> 00:03:45,270 We leave the X here like this. 56 00:03:45,360 --> 00:03:52,190 We add one to the exponent so we get 0 plus 1 and we divide by our new exponent. 57 00:03:52,200 --> 00:03:59,430 So we say 0 plus 1 here and you'll see how that's going to simplify to just X like we saw before. 58 00:03:59,670 --> 00:04:04,440 So now we simplify this we get x cubed plus two divided by two is 1. 59 00:04:04,440 --> 00:04:05,930 So that's going to disappear. 60 00:04:06,090 --> 00:04:09,300 We get X squared plus x squared. 61 00:04:09,450 --> 00:04:13,170 And then here zero plus 1 is 1 1 divided by one is one. 62 00:04:13,170 --> 00:04:15,090 So this will disappear also. 63 00:04:15,360 --> 00:04:19,500 And we're left with X to the first power so we just have plus X like this. 64 00:04:19,500 --> 00:04:26,280 And remember we saw that here when we took the integral of one we got 1 x or just X and you can see 65 00:04:26,280 --> 00:04:33,480 we ended up with X here so that's technically why the integral of a constant is just that constant multiplied 66 00:04:33,510 --> 00:04:34,930 by x. 67 00:04:35,160 --> 00:04:42,150 So this is our indefinite integral except that our last step is to add c whenever you're dealing with 68 00:04:42,150 --> 00:04:43,570 an indefinite integral. 69 00:04:43,680 --> 00:04:47,190 You always have to add c which we call the constant of integration. 70 00:04:47,190 --> 00:04:52,920 The reason that we have to add that is because there are several possible answers for the integral of 71 00:04:52,920 --> 00:04:54,840 this particular function. 72 00:04:54,840 --> 00:05:01,080 The reason is because of this constant value here or any constant value remember that when you take 73 00:05:01,080 --> 00:05:04,530 the derivative of a constant the derivative is zero. 74 00:05:04,530 --> 00:05:04,830 Right. 75 00:05:04,830 --> 00:05:11,930 If we take the derivative of one we get zero if we take the derivative of four we get zero. 76 00:05:11,940 --> 00:05:14,910 If we take the derivative of Pi we get zero. 77 00:05:14,910 --> 00:05:17,780 The derivative of any constant is zero. 78 00:05:17,790 --> 00:05:25,080 So when we're reversing the process we don't know if the original function had some extra constant added 79 00:05:25,080 --> 00:05:25,810 onto it. 80 00:05:25,860 --> 00:05:31,340 Our original function could have been x cubed plus x squared plus X plus 4. 81 00:05:31,410 --> 00:05:36,930 And if it was when we took its derivative we would have ended up with this value here 3 x squared plus 82 00:05:36,930 --> 00:05:38,120 2 x plus 1. 83 00:05:38,220 --> 00:05:42,950 But there would have been no sign of the constant the constant would have just disappeared. 84 00:05:42,960 --> 00:05:49,140 So this polynomial here has to include this plus C value because what it tells us is that the integral 85 00:05:49,140 --> 00:05:55,800 of this original function here is this polynomial Plus it could have had some constant attached to it. 86 00:05:55,830 --> 00:05:58,900 Any of these constants for maybe just 0. 87 00:05:58,920 --> 00:06:03,330 But either way we have to account for the possibility that there was a constant here so whenever we 88 00:06:03,330 --> 00:06:07,530 have this indefinite integral we always have to add the constant of integration. 89 00:06:07,560 --> 00:06:10,540 Once we do we can say that this is our final answer. 9670