Antiderivatives - Free Educational videos for Students in K-12

Antiderivatives - By The Organic Chemistry Tutor

Transcript


00:00	in this video , we're going to talk about how
00:02	to find the anti derivative of a function . So
00:05	what is the anti derivative ? What as the name
00:10	implies ? It is the opposite of differentiation . We
00:14	know the derivative of F of X is F prime
00:18	of X . So the anti derivative or the integral
00:22	of F prime of X is F of X .
00:27	So F of X is the anti derivative of F
00:30	prime of X where F prime of X is the
00:33	derivative of F of X . Now the integral or
00:39	the anti derivative of regular F of X is capital
00:45	F of X . So capital F is the anti
00:51	derivative of lowercase F . And lower case F is
00:55	the anti derivative of F prime of X . So
00:58	hopefully that gives you a good idea of what the
01:00	anti derivative is . So let's say if we have
01:08	the second derivative of a function and we want to
01:11	find the first derivative and we can go we can
01:16	keep going forward in this direction . I just want
01:20	to give you an overview of these different symbols and
01:24	functions . So go into the right , we're looking
01:29	for the anti derivative , going to the left ,
01:39	we're looking for the derivative . The process of finding
01:47	the anti derivative is basically integration . The process of
01:55	finding the derivative is differentiation . Hopefully that summarizes everything
02:07	that you need to know about this conceptually . No
02:15	, we said that capital F is the integral of
02:19	F . Of X . Dx . Now you you
02:25	get capital F plus C . You always get the
02:27	a constant integration but we're not gonna worry about that
02:30	here . What I'd like to do right now is
02:33	distinguish what is an indefinite integral versus a definite integral
02:43	. So this here is what is known as an
02:47	indefinite integral . An indefinite integral gives you a function
02:54	typically in terms of X . But it could be
02:56	in terms of some other variable like Y or Z
02:58	. Or something like that . A definite integral doesn't
03:03	give you a function , but it gives you a
03:06	number . So this is a definite integral . The
03:15	result of a definite integral . It's always a number
03:17	like 20-845 or something . And you can easily identify
03:21	or distinguish a definite integral from an indefinite integral .
03:26	A definite integral have limits of integration , it has
03:29	a lower limit and the upper limit an indefinite integral
03:35	does not have the limits of integration . So that's
03:39	how you can easily distinguish between them . Now we
03:43	know that the derivative of X cube is three x
03:51	squared according to the power rule . The derivative of
03:56	a variable race to a constant is the constant times
04:01	the variable , erase the constant minus one . Well
04:06	, the anti derivative of a variable race to a
04:11	constant . It's going to be that variable , raising
04:14	the constant plus one Divided by the constant plus one
04:19	and then plus see the constant of integration . So
04:24	that's how we could find the anti derivative of a
04:27	variable race to a constant . So let's find the
04:32	anti derivative of three X squared . So let's rewrite
04:37	the constant . Now the variable rates of the constant
04:40	that's X squared . So we're gonna do is we're
04:42	gonna add one to the exponent . So it's gonna
04:45	be two plus one and then we're gonna divide by
04:47	that result . So it becomes three x cubed divided
04:51	by three . And this gives us the original function
04:57	X cube . But we need to add the constant
05:00	of integration plus C . If you were to find
05:04	the derivative of X cubed plus C . X cube
05:10	will become three X squared . The derivative of a
05:14	constant will become zero . So you end up with
05:16	the same thing . So always when you're looking for
05:19	the indefinite integral something or the anti derivative , you
05:23	need to put the constant of integration . The only
05:26	time you don't need to worry about the constant of
05:28	integration is if you're finding the definite integral where you
05:31	get a number besides that , any time you get
05:35	a function add plus C . When you're looking for
05:37	the indefinite integral or the anti derivative . So now
05:43	let's work on some problems , go ahead and find
05:48	the anti derivative of the following functions . Feel free
05:56	to pause the video and work on these examples .
06:07	So for the first one we're just going to add
06:10	one to the exponent four plus one is five .
06:13	And then we're going to divide by the result and
06:15	then add the constant of integration . So it's X
06:18	to the fifth power over five plus C . For
06:21	the next one is going to be X to the
06:25	8/8 plus c . For the next one is gonna
06:29	be six . And then we're going to add 1-9
06:31	that becomes 10 , Divide by 10 Plus C .
06:35	Now this one we can reduce 6/10 . If we
06:38	divide both numbers by two , that becomes three of
06:40	the five . So we could say this equals 3/5
06:45	. Exit a 10 Plus C . For the next
06:52	one It's gonna be eight X to the 4/4 plus
06:57	c . And we know eight divided by four is
07:00	two . So this is two X to the four
07:03	plus C . Sat . You can see it's not
07:07	that difficult to find the anti derivative of function .
07:14	So go ahead and try this one . Find the
07:17	anti derivative of . So let's try it this way
07:21	let's say F of X is X cubed minus four
07:28	X squared plus eight X . What is the anti
07:34	derivative capital F of X of their function . So
07:39	go ahead and find the anti drift F of X
07:44	. To do that . We're going to find the
07:45	indefinite integral of X cubed minus four x squared plus
07:51	attacks . But just to show our work , let's
07:55	do it this way first . So the anti derivative
07:59	is going to be the indefinite integral of F of
08:02	X dx . So that's the formula . And now
08:08	let's replace F of X with xq minus four X
08:13	squared Plus eight X DX . So the anti derivative
08:24	of X cube is going to be X to the
08:26	fore over four . And then for four X squared
08:31	is going to be four X cube over three .
08:33	And then for a tax it's eight X squared over
08:35	two plus C . Now let's rewrite it . So
08:40	we're gonna be right as 1 4th except for -4/3
08:47	X cube and then 8/2 is four . So we're
08:51	right that has four X squared plus C . So
08:54	this is the anti derivative of the original function .
09:04	Now let's try another problem . So let's say F
09:08	prime of X . That is the first derivative of
09:11	X . I mean the first derivative of F .
09:14	Rather let's say that's equal to eight X cube -6
09:26	x squared plus four X minus seven . So given
09:34	F prime of X , find the anti derivative F
09:38	of X . Go ahead and try that . So
09:51	, first let's right an expression involving indefinite integral .
09:54	So we can say that F of X is the
09:56	indefinite integral of F prime of X . Dx So
10:03	F of X is the anti derivative of F prime
10:05	of X . So f of X is going to
10:11	be the indefinite integral of eight . X cube -6
10:18	x squared plus four X minus seven . Feel free
10:28	to pause the video if you want to try this
10:38	. So the anti derivative of X cube is X
10:43	to the 4th over four . For X squared it's
10:47	X Cube over three and for X to the first
10:51	power It's x squared over two . And then if
10:55	you have a constant like seven , all we gotta
10:57	do is added next to it . We'll talk more
11:02	about that shortly and then finally plus the So now
11:07	let's simplify our answer . So the anti derivative is
11:11	going to be eight divided by four is two .
11:14	So we have two X to the fourth , power
11:17	six divided by three is also too four divided by
11:20	two is the same . So this is the anti
11:26	derivative of our original function two X to the fourth
11:29	power minus two X cubed plus two X squared -7
11:35	X Plus C . So , as you can see
11:39	it's not too difficult to find an anti derivative ,
11:41	but there are other things that you need to know
11:45	. Go ahead and find the indefinite integral of these
11:50	expressions . Let's make this as easy . All right
12:18	, so go ahead and try those . So let's
12:22	start with the first one . We want to find
12:24	the indefinite integral of a constant five and we have
12:28	dx in front of it . So we know it's
12:31	going to be five x . But let's see if
12:34	we can get the answer . Using the power role
12:36	five is the same as five . Exit zero .
12:40	Any variable rates to the zero , power is one
12:43	. So excited Zeros one times 5 , Which is
12:47	equivalent to five . Now using the power rule ,
12:51	this is going to be five . We're going to
12:53	add on to the exponents and then divide by that
12:55	result . So the end result is that you get
12:58	five X plus C . So whenever you have a
13:04	constant and you wish to integrate that constant , just
13:08	add a variable to it . Now the variable that
13:10	you should add is based on what you see here
13:13	. Either dx Dy , D R or D .
13:17	C . So what is the indefinite integral of 70
13:21	y . So instead of seven X this is going
13:24	to be seven times Y plus C . So the
13:32	differential dx dy it tells us what invariable we should
13:36	be adding to the constants . The indefinite integral of
13:41	eight D . R . That's going to be eight
13:43	times are plus C . For six z plus four
13:48	. It's gonna be six . And then Z to
13:51	the first power that becomes Z squared over two ,
13:54	four becomes foresee and then we'll add plus C .
13:58	So this we could simplify it as three Z squared
14:03	Plus four C Plus C . So that is the
14:07	indefinite integral of 60 plus four . Try this problem
14:16	. So let's say you're given the first derivative of
14:20	F . And let's say it's seven . This should
14:26	be the F F expert F . Prime of Y
14:32	. So let's say it's seven y cube -3 Y
14:37	Plus eight . And your goal is to find capital
14:43	F of Y . How would you do it ?
14:46	Feel free to pause the video and try it .
14:50	So first let's vita mental outline of what we need
14:54	to do here were given the first derivative of F
14:59	. We need to find F . So F is
15:03	the anti derivative of F . Prime , but we're
15:05	not going to stop there . We need to find
15:06	capital F . So what we need to do is
15:11	we need to integrate the function two times . So
15:15	we have a successive integration problem . So first let's
15:23	integrate it the first time , let's find regular F
15:26	. Or little F . So this is going to
15:29	be the indefinite integral of F . Prime of Y
15:32	. Dy . So that is the indefinite integral of
15:37	seven . Y cube -3 Y Plus eight . So
15:46	the anti derivative of Y to the third is going
15:48	to be Y to the 4th over four . The
15:51	anti derivative of why His wife squared over two .
15:55	And for the constant eight , there's just gonna be
15:57	eight Y based on what we have here in the
16:00	UAE and then plus the constancy . So that's F
16:06	. Of Y . We don't know what the value
16:08	of C . Is . If we were given a
16:12	point of F . F . Y , we could
16:15	find see . But we'll save that for another problem
16:18	. So this here is the general function of the
16:23	movie . So now let's find capital F . Of
16:28	Y . That's going to be the indefinite integral of
16:35	F . Of Y . Dy . That is the
16:38	indefinite integral of 7/4 . Why ? to the 4th
16:45	-3/2 ? Y squared plus eight Y plus C .
16:51	As well . And we're gonna have . Dy so
17:14	the anti derivative of this , we're going to rewrite
17:17	the constant And then we'll find the anti derivative of
17:20	Y . to the 4th . So that's going to
17:22	be why to the 5th over five . And then
17:26	minus let's rewrite the constant first . So we have
17:30	3/2 . And then the anti derivative of Y squared
17:33	is going to be Y to the third over three
17:37	. And then it's going to be plus eight times
17:39	Y squared over two . Now we have the constant
17:43	of integration . See , So we're going to add
17:46	a Y . To it . It's gonna be C
17:49	. Times Y . And then plus a new constant
17:51	which will call D . So this becomes seven over
18:03	20 . Why ? to the 5th here we can
18:08	cancel the threes . So it's going to be -1
18:11	half . What's the third plus four ? Y squared
18:17	plus C . Y plus T . So that is
18:24	the second anti derivative of F . Prime of Y
18:29	. Which is capital F . Of life by the
18:32	way . For those of you who want access to
18:34	the full length version of this video . Check out
18:37	the links in the description section below . If you
18:41	join my Youtube channel membership program , you can get
18:43	access not only to the full length version of this
18:46	video , but many other full length versions of other
18:50	videos that I have on Youtube which you'll see over
18:53	time . So feel free to take a look at
18:56	that when you get a chance . Now , let's
18:58	work on finding the indefinite integral of other types of
19:01	functions . What is the indefinite integral of pie D
19:07	theater ? Feel free to pause the video and work
19:15	on that . What is the indefinite integral of E
19:20	let's say E diaz . So pi is a constant
19:33	. Let's compare to the indefinite integral of seven .
19:39	Dy seven is a constant and we would add the
19:43	variable Y to it . So this will be seven
19:45	Y plus C . So pi is a constant .
19:48	And here we're going to add the variable fada to
19:50	it plus C . So that's the indefinite integral of
19:54	pi D . Data . So remember , pi is
19:59	a constant , is 3.14 with some other numbers after
20:03	that . Let me find out what those other numbers
20:07	are . So it's 3.14 159 2654 . With some
20:14	other numbers . So whenever you see pie , it's
20:17	just a constant . So treat it as if it's
20:21	any of the number like seven or 8 or -4
20:24	. The same is true for E . He is
20:26	not a variable , it's a constant . E .
20:29	is equal to 2.7 1828 and so forth . So
20:39	when looking at a problem like this is the constant
20:41	, as is the variable . So this is going
20:44	to be easy times , S plus C . Whenever
20:49	you're finding the indefinite integral of something , you need
20:52	to be able to distinguish the constant from the variable
20:55	. In order to integrate it properly . Now let's
21:01	focus on finding the indefinite integral of rational functions .
21:05	What is the indefinite integral of one over X squared
21:11	. To find this before you integrated , you need
21:13	to rewrite the expression , we need to move the
21:16	variable from the bottom to the top . So this
21:20	is X . To the negative too . At this
21:24	point we could use the power rule so we can
21:27	integrate it by adding one to the exponents . So
21:30	it's gonna be negative two plus one . And then
21:33	divided by that result plus C . -2 Plus one
21:39	. That's -1 . And after you integrate it you
21:44	need to rewrite it again . So we want to
21:46	get rid of this negative exploring and we can do
21:48	so by moving the variable back to the bottom .
21:54	So that leaves a one on top . And then
21:56	we're gonna have X . To the first power on
21:58	the bottom which we can just leave us X .
22:01	The negative sign . I'm going to move it to
22:02	the top . So the final answer is negative one
22:06	over X . Plus C . Now let's try this
22:17	one . What is the indefinite integral of one over
22:20	X . To the fifth power dx . So feel
22:25	free to pause the video and try it . So
22:28	just like before let's move the variable to the top
22:32	. So we have the integral of X . Race
22:36	the -5 detox . And then let's add 1 to
22:39	that . So negative five plus one is negative four
22:43	. And then we're going to divide by that result
22:46	. Next let's rewrite the expression . So we're going
22:49	to move the variable back to the bottom and the
22:52	negative sign . We're going to move it to the
22:53	top . So we're gonna have negative one . We
22:58	have a four on the bottom and X . And
23:01	then this negative four becomes positive for plus C .
23:05	So negative 1/4 . X . To the fourth plus
23:07	C . That is the indefinite integral Of one over
23:12	X to the 5th power . Try this . Finally
23:18	indefinite integral of seven over X cube dx . So
23:27	like before we're going to move the variable to the
23:29	top . So we're going to rewrite The integral .
23:33	So this is seven X . to the negative three
23:37	detox . And that was integrated . So negative three
23:42	plus one . That's negative two . And then we're
23:46	gonna divide by negative two and then we'll have plus
23:48	C . So next we're going to move the variable
23:52	back to the bottom and the negative sign . I'm
23:54	going to move to the top . So this becomes
23:57	-7 . We have two on the bottom and we
24:03	have X squared as well . So it's negative seven
24:07	over two X squared plus C . So that's how
24:10	you can integrate rational functions . Go ahead and try
24:20	this expression Finally indefinite integral of five . Exodus seven
24:26	-9 over x squared plus four x -8 dx .
24:36	So this is going to be five exit eight over
24:40	eight . Now for negative non X squared , we
24:43	need to write that as -9 X . to the
24:46	negative too . So that's gonna be negative nine X
24:50	. Negative two plus one is negative one divided by
24:53	negative one . And then for four X . To
24:55	the first power , that will be four x squared
24:58	over two and then minus attacks plus C . So
25:04	we can rewrite this as 5/8 . Exit eight plus
25:11	nine . The two negative signs will cancel and then
25:14	we can move the X variable to the bottom .
25:19	So it's plus , whoops , It's plus nine over
25:24	X . And then plus two X squared -8 X
25:32	. Policy . So that is the indefinite integral or
25:36	the anti derivative of this expression . Now what is
25:46	the anti derivative of one of her ex Zodiacs ?
25:51	What do you think we need to do here ?
25:54	Well , if we were to try the same approach
25:56	that we've been using for rational functions , let's say
26:00	if we were to rewrite the expression by moving X
26:03	to the top , We would get the indefinite integral
26:05	of X to the -1 . Detox . Now using
26:10	the power rule , if we add 12 -1 will
26:13	get excited zero . And if we were to divide
26:16	by that result , we would get something that is
26:20	undefined . So this technique doesn't work for one of
26:24	the acts . Instead , this is an integral that
26:28	you should commit to memory . The integral of one
26:31	of the X . Dx is simply L . N
26:34	. X . To me , I want to add
26:36	that to your list of notes . And if you
26:41	recall the derivative of Ellen you where you as a
26:49	function of X . Is you prime over you ?
26:54	So the derivative of Ln X . It's gonna be
26:59	you prime . If you S X . You prime
27:03	the derivative exes one . So we get one over
27:06	x . So that's the anti derivative of one over
27:10	X . To the first power . So if the
27:15	anti derivative of one of her access L N X
27:18	. What is the anti derivative of one over X
27:22	plus three ? This is simply the natural log of
27:30	X plus tree . So based on that , go
27:44	ahead and find the anti derivative of the following expressions
27:49	one over X -5 DX . Five over X plus
27:57	two dx and seven over X plus four dx .
28:07	So this is going to be the natural log Of
28:14	X -5 Plus C . Now typically if your answer
28:20	is like different than L . N . X .
28:22	Normally you'll see an absolute value around X -5 .
28:26	And the reason for this is that you can have
28:29	a negative number inside the natural log , it's gonna
28:31	be like it's going to give you an error in
28:34	the calculator . So typically you'll see an absolute value
28:38	symbol for expressions that can be negative For the next
28:44	one . Let's move the constant to the front .
28:48	So think of this as five times the integral of
28:53	one over X plus two D . Ax . So
28:56	this is going to be five times the natural log
29:00	. And let's put our absolute value symbol . So
29:04	it's five alan X plus two plus the constant of
29:11	integration for the next one . Let's do the same
29:15	thing . Let's move the constant to the front .
29:20	So this is going to be seven natural log X
29:28	plus four and then plus C . So that's how
29:34	you can integrate rational functions . Where the denominator is
29:38	a linear function like X plus two , x minus
29:40	five and so forth . Now here's a general formula
29:48	that you can use when integrating rational functions . Like
29:52	one of you will use the function of X .
29:56	It's gonna be one over you , prime times the
30:03	natural log of you plus C . Now this works
30:11	if this is a big if if you is a
30:14	linear function in the form A X plus B .
30:18	Now it could be zero . BkB zero , Wolf
30:22	, a zero doesn't work , but BkB zero .
30:27	But you have to be a linear function in that
30:29	form . If there's like an X squared or an
30:31	X cube , this formula is not gonna work .
30:34	So let me repeat that . You have to be
30:36	linear function if you have a square root of X
30:39	or anything , but exit the first power . This
30:42	will not work . So now let's try some harder
30:46	examples Based on that . What is the indefinite integral
30:50	of one Over three Acts Plus 4 ? And let's
30:55	compare that to the indefinite integral of one over X
30:59	plus seven . So in this case we can see
31:04	that you is three x plus four the derivative of
31:09	you which is you crime . The derivative of three
31:13	accessory . The derivative of four is just nothing .
31:17	See you prime mystery . So this becomes 1/3 .
31:20	Ellen absolute value three acts plus four and then plus
31:33	city . So this fraction is a result of the
31:39	derivative of that expression . For the next one the
31:44	derivative of X plus seven inches one . He had
31:47	the derivative of three x plus four stream . So
31:50	that's why you got to incorporate that . So for
31:53	the next one you is going to be X plus
31:55	seven and you prime . It's just going to be
31:58	one . So this becomes Ellen absolute value X plus
32:04	seven plus C . So no one that go ahead
32:14	and try these two . Oh by the way ,
32:19	let's go back to what we had before . Let's
32:22	prove that this is indeed the right answer . So
32:30	let's find the derivative of one third times Ln three
32:38	x plus four . So this is going to be
32:44	won over three . And keep in mind the derivative
32:48	of L N X . Or Ellen you rather is
32:53	you prime over you . So the you part is
32:56	what is inside of the natural log function . And
33:00	so that's three X plus for you prime the derivative
33:04	of three X plus four is history And the derivative
33:08	of C . is zero . And so when we
33:12	cancel the threes , we're going to start with our
33:16	original function , which is what we have here .
33:21	So that shows you that the process works .

Summarizer

DESCRIPTION:

This calculus video tutorial provides a basic introduction into antiderivatives. It explains how to find the indefinite integral of polynomial functions as well as rational functions.

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Antiderivatives is a free educational video by The Organic Chemistry Tutor.

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