[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.58,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.58,0:00:02.26,Default,,0000,0000,0000,,Let's take the\Nindefinite integral
Dialogue: 0,0:00:02.26,0:00:07.99,Default,,0000,0000,0000,,of the square root\Nof 7x plus 9 dx.
Dialogue: 0,0:00:07.99,0:00:10.29,Default,,0000,0000,0000,,So my first question\Nto you is, is this
Dialogue: 0,0:00:10.29,0:00:14.06,Default,,0000,0000,0000,,going to be a good case\Nfor u-substitution?
Dialogue: 0,0:00:14.06,0:00:16.83,Default,,0000,0000,0000,,Well, when you look here,\Nmaybe the natural thing
Dialogue: 0,0:00:16.83,0:00:20.88,Default,,0000,0000,0000,,to set to be equal\Nto u is 7x plus 9.
Dialogue: 0,0:00:20.88,0:00:24.20,Default,,0000,0000,0000,,But do I see its derivative\Nanywhere over here?
Dialogue: 0,0:00:24.20,0:00:24.87,Default,,0000,0000,0000,,Well, let's see.
Dialogue: 0,0:00:24.87,0:00:30.12,Default,,0000,0000,0000,,If we set u to be\Nequal to 7x plus 9,
Dialogue: 0,0:00:30.12,0:00:33.41,Default,,0000,0000,0000,,what is the derivative of u\Nwith respect to x going to be?
Dialogue: 0,0:00:33.41,0:00:35.31,Default,,0000,0000,0000,,Derivative of u\Nwith respect to x
Dialogue: 0,0:00:35.31,0:00:37.08,Default,,0000,0000,0000,,is just going to be equal to 7.
Dialogue: 0,0:00:37.08,0:00:38.47,Default,,0000,0000,0000,,Derivative of 7x is 7.
Dialogue: 0,0:00:38.47,0:00:40.65,Default,,0000,0000,0000,,Derivative of 9 is 0.
Dialogue: 0,0:00:40.65,0:00:44.19,Default,,0000,0000,0000,,So do we see a 7 lying\Naround anywhere over here?
Dialogue: 0,0:00:44.19,0:00:45.58,Default,,0000,0000,0000,,Well, we don't.
Dialogue: 0,0:00:45.58,0:00:49.34,Default,,0000,0000,0000,,But what could we do in order\Nto have a 7 lying around,
Dialogue: 0,0:00:49.34,0:00:53.28,Default,,0000,0000,0000,,but not change the\Nvalue of the integral?
Dialogue: 0,0:00:53.28,0:00:55.96,Default,,0000,0000,0000,,Well, the neat thing-- and we've\Nseen this multiple times-- is
Dialogue: 0,0:00:55.96,0:00:57.33,Default,,0000,0000,0000,,when you're\Nevaluating integrals,
Dialogue: 0,0:00:57.33,0:01:01.24,Default,,0000,0000,0000,,scalars can go in and outside\Nof the integral very easily.
Dialogue: 0,0:01:01.24,0:01:05.92,Default,,0000,0000,0000,,Just to remind ourselves, if I\Nhave the integral of let's say
Dialogue: 0,0:01:05.92,0:01:11.69,Default,,0000,0000,0000,,some scalar a times\Nf of x dx, this
Dialogue: 0,0:01:11.69,0:01:17.42,Default,,0000,0000,0000,,is the same thing as a times\Nthe integral of f of x dx.
Dialogue: 0,0:01:17.42,0:01:19.34,Default,,0000,0000,0000,,The integral of the\Nscalar times a function
Dialogue: 0,0:01:19.34,0:01:22.65,Default,,0000,0000,0000,,is equal to the scalar times\Nthe integral of the functions.
Dialogue: 0,0:01:22.65,0:01:25.28,Default,,0000,0000,0000,,So let me put this\Naside right over here.
Dialogue: 0,0:01:25.28,0:01:28.63,Default,,0000,0000,0000,,So with that in mind, can\Nwe multiply and divide
Dialogue: 0,0:01:28.63,0:01:31.71,Default,,0000,0000,0000,,by something that will\Nhave a 7 showing up?
Dialogue: 0,0:01:31.71,0:01:34.33,Default,,0000,0000,0000,,Well, we can multiply\Nand divide by 7.
Dialogue: 0,0:01:34.33,0:01:35.83,Default,,0000,0000,0000,,So imagine doing this.
Dialogue: 0,0:01:35.83,0:01:38.58,Default,,0000,0000,0000,,Let's rewrite our\Noriginal integral.
Dialogue: 0,0:01:38.58,0:01:40.96,Default,,0000,0000,0000,,So let me draw a\Nlittle arrow here just
Dialogue: 0,0:01:40.96,0:01:42.42,Default,,0000,0000,0000,,to go around that aside.
Dialogue: 0,0:01:42.42,0:01:44.10,Default,,0000,0000,0000,,We could rewrite our\Noriginal integral
Dialogue: 0,0:01:44.10,0:01:51.34,Default,,0000,0000,0000,,as being 9 to the integral\Nof times 1/7 times 7 times
Dialogue: 0,0:01:51.34,0:01:57.98,Default,,0000,0000,0000,,the square root of 7x plus 9 dx.
Dialogue: 0,0:01:57.98,0:02:00.06,Default,,0000,0000,0000,,And if we want to, we\Ncould take the 1/7 outside
Dialogue: 0,0:02:00.06,0:02:00.73,Default,,0000,0000,0000,,of the integral.
Dialogue: 0,0:02:00.73,0:02:02.27,Default,,0000,0000,0000,,We don't have to,\Nbut we can rewrite
Dialogue: 0,0:02:02.27,0:02:06.62,Default,,0000,0000,0000,,this as 1/7 times\Nthe integral of 7,
Dialogue: 0,0:02:06.62,0:02:12.13,Default,,0000,0000,0000,,times the square\Nroot of 7x plus 9 dx.
Dialogue: 0,0:02:12.13,0:02:14.84,Default,,0000,0000,0000,,So now if we set u\Nequal to 7x plus 9,
Dialogue: 0,0:02:14.84,0:02:16.86,Default,,0000,0000,0000,,do we have its\Nderivative laying around?
Dialogue: 0,0:02:16.86,0:02:17.41,Default,,0000,0000,0000,,Well, sure.
Dialogue: 0,0:02:17.41,0:02:20.35,Default,,0000,0000,0000,,The 7 is right over here.
Dialogue: 0,0:02:20.35,0:02:23.00,Default,,0000,0000,0000,,We know that du-- if we want to\Nwrite it in differential form--
Dialogue: 0,0:02:23.00,0:02:27.06,Default,,0000,0000,0000,,du is equal to 7 times dx.
Dialogue: 0,0:02:27.06,0:02:31.46,Default,,0000,0000,0000,,So du is equal to 7 times dx.
Dialogue: 0,0:02:31.46,0:02:35.32,Default,,0000,0000,0000,,That part right over\Nthere is equal to du.
Dialogue: 0,0:02:35.32,0:02:37.32,Default,,0000,0000,0000,,And if we want to care\Nabout u, well, that's
Dialogue: 0,0:02:37.32,0:02:40.05,Default,,0000,0000,0000,,just going to be the 7x plus 9.
Dialogue: 0,0:02:40.05,0:02:41.52,Default,,0000,0000,0000,,That is are u.
Dialogue: 0,0:02:41.52,0:02:45.27,Default,,0000,0000,0000,,So let's rewrite this indefinite\Nintegral in terms of u.
Dialogue: 0,0:02:45.27,0:02:53.12,Default,,0000,0000,0000,,It's going to be equal to\N1/7 times the integral of--
Dialogue: 0,0:02:53.12,0:02:55.16,Default,,0000,0000,0000,,and I'll just take the 7\Nand put it in the back.
Dialogue: 0,0:02:55.16,0:02:57.71,Default,,0000,0000,0000,,So we could just\Nwrite the square root
Dialogue: 0,0:02:57.71,0:03:06.45,Default,,0000,0000,0000,,of u du, 7 times dx is du.
Dialogue: 0,0:03:06.45,0:03:10.11,Default,,0000,0000,0000,,And we can rewrite this if we\Nwant as u to the 1/2 power.
Dialogue: 0,0:03:10.11,0:03:12.32,Default,,0000,0000,0000,,It makes it a little bit\Neasier for us to kind of do
Dialogue: 0,0:03:12.32,0:03:14.49,Default,,0000,0000,0000,,the reverse power rule here.
Dialogue: 0,0:03:14.49,0:03:20.94,Default,,0000,0000,0000,,So we can rewrite this as equal\Nto 1/7 times the integral of u
Dialogue: 0,0:03:20.94,0:03:23.75,Default,,0000,0000,0000,,to the 1/2 power du.
Dialogue: 0,0:03:23.75,0:03:25.00,Default,,0000,0000,0000,,And let me just make it clear.
Dialogue: 0,0:03:25.00,0:03:26.68,Default,,0000,0000,0000,,This u I could have\Nwritten in white
Dialogue: 0,0:03:26.68,0:03:27.96,Default,,0000,0000,0000,,if I want it the same color.
Dialogue: 0,0:03:27.96,0:03:31.10,Default,,0000,0000,0000,,And this du is the same\Ndu right over here.
Dialogue: 0,0:03:31.10,0:03:35.79,Default,,0000,0000,0000,,So what is the antiderivative\Nof u to the 1/2 power?
Dialogue: 0,0:03:35.79,0:03:39.05,Default,,0000,0000,0000,,Well, we increment\Nu's power by 1.
Dialogue: 0,0:03:39.05,0:03:41.01,Default,,0000,0000,0000,,So this is going to be\Nequal to-- let me not
Dialogue: 0,0:03:41.01,0:03:43.26,Default,,0000,0000,0000,,forget this 1/7 out front.
Dialogue: 0,0:03:43.26,0:03:49.11,Default,,0000,0000,0000,,So it's going to be 1/7 times--\Nif we increment the power here,
Dialogue: 0,0:03:49.11,0:03:55.52,Default,,0000,0000,0000,,it's going to be u to the 3/2,\N1/2 plus 1 is 1 and 1/2 or 3/2.
Dialogue: 0,0:03:55.52,0:03:57.28,Default,,0000,0000,0000,,So it's going to\Nbe u to the 3/2.
Dialogue: 0,0:03:57.28,0:04:01.03,Default,,0000,0000,0000,,
Dialogue: 0,0:04:01.03,0:04:04.14,Default,,0000,0000,0000,,And then we're going to\Nmultiply this new thing
Dialogue: 0,0:04:04.14,0:04:07.98,Default,,0000,0000,0000,,times the reciprocal\Nof 3/2, which is 2/3.
Dialogue: 0,0:04:07.98,0:04:10.84,Default,,0000,0000,0000,,And I encourage you to verify\Nthe derivative of 2/3 u
Dialogue: 0,0:04:10.84,0:04:14.63,Default,,0000,0000,0000,,to the 3/2 is\Nindeed u to the 1/2.
Dialogue: 0,0:04:14.63,0:04:15.75,Default,,0000,0000,0000,,And so we have that.
Dialogue: 0,0:04:15.75,0:04:17.45,Default,,0000,0000,0000,,And since we're\Nmultiplying 1/7 times
Dialogue: 0,0:04:17.45,0:04:19.44,Default,,0000,0000,0000,,this entire indefinite\Nintegral, we
Dialogue: 0,0:04:19.44,0:04:21.56,Default,,0000,0000,0000,,could also throw in a\Nplus c right over here.
Dialogue: 0,0:04:21.56,0:04:23.43,Default,,0000,0000,0000,,There might have\Nbeen a constant.
Dialogue: 0,0:04:23.43,0:04:25.99,Default,,0000,0000,0000,,And if we want, we can\Ndistribute the 1/7.
Dialogue: 0,0:04:25.99,0:04:36.17,Default,,0000,0000,0000,,So it would get 1/7 times\N2/3 is 2/21 u to the 3/2.
Dialogue: 0,0:04:36.17,0:04:38.75,Default,,0000,0000,0000,,And 1/7 times some\Nconstant, well, that's
Dialogue: 0,0:04:38.75,0:04:40.21,Default,,0000,0000,0000,,just going to be some constant.
Dialogue: 0,0:04:40.21,0:04:41.96,Default,,0000,0000,0000,,And so I could write\Na constant like that.
Dialogue: 0,0:04:41.96,0:04:44.51,Default,,0000,0000,0000,,I could call that c1 and\Nthen I could call this c2,
Dialogue: 0,0:04:44.51,0:04:47.14,Default,,0000,0000,0000,,but it's really just\Nsome arbitrary constant.
Dialogue: 0,0:04:47.14,0:04:47.85,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:04:47.85,0:04:49.18,Default,,0000,0000,0000,,Oh, actually, no we aren't done.
Dialogue: 0,0:04:49.18,0:04:51.52,Default,,0000,0000,0000,,We still just have our\Nentire thing in terms of u.
Dialogue: 0,0:04:51.52,0:04:54.29,Default,,0000,0000,0000,,So now let's unsubstitute it.
Dialogue: 0,0:04:54.29,0:05:01.29,Default,,0000,0000,0000,,So this is going to be equal\Nto 2/21 times u to the 3/2.
Dialogue: 0,0:05:01.29,0:05:03.71,Default,,0000,0000,0000,,And we already know\Nwhat u is equal to.
Dialogue: 0,0:05:03.71,0:05:06.06,Default,,0000,0000,0000,,u is equal to 7x plus 9.
Dialogue: 0,0:05:06.06,0:05:08.72,Default,,0000,0000,0000,,Let me put a new color here\Njust to ease the monotony.
Dialogue: 0,0:05:08.72,0:05:13.46,Default,,0000,0000,0000,,So it's going to be\N2/21 times 7x plus 9
Dialogue: 0,0:05:13.46,0:05:18.77,Default,,0000,0000,0000,,to the 3/2 power plus c.
Dialogue: 0,0:05:18.77,0:05:21.87,Default,,0000,0000,0000,,
Dialogue: 0,0:05:21.87,0:05:23.03,Default,,0000,0000,0000,,And we are done.
Dialogue: 0,0:05:23.03,0:05:25.50,Default,,0000,0000,0000,,We were able to take a kind\Nof hairy looking integral
Dialogue: 0,0:05:25.50,0:05:28.12,Default,,0000,0000,0000,,and realize that even\Nthough it wasn't completely
Dialogue: 0,0:05:28.12,0:05:31.87,Default,,0000,0000,0000,,obvious at first, that\Nu-substitution is applicable.
Dialogue: 0,0:05:31.87,0:05:32.37,Default,,0000,0000,0000,,