Indefinite and definite integrals
usubstitution
None
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(2^(ln x))/x antiderivative example
Finding ∫(2^ln x)/x dx
Discussion and questions for this video
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 I was just looking on the discussion boards on the Khan
 Academy Facebook page, and Bud Denny put up this problem,
 asking for it to be solved.
 And it seems like a problem of general interest.
 If the indefinite integral of 2 to the natural log of x
 over, everything over x, dx.
 And on the message board, Abhi Khanna also put up a solution,
 and it is the correct solution, but I thought this was of
 general interest, so I'll make a quick video on it.
 So the first thing when you see an integral like this, is you
 say, hey, you know, I have this natural log of x up in
 the numerator, and where do I start?
 And the first thing that should maybe pop out at you, is that
 this is the same thing as the integral of one over x times 2
 to the natural log of x, dx.
 And so you have an expression here, or it's kind of part of
 our larger function, and you have its derivative, right?
 We know that the derivative, let me write it over here, we
 know that the derivative with respect to x of the natural
 log of x is equal to 1/x.
 So we have some expression, and we have its derivative, which
 tells us that we can use substitution.
 Sometimes you can do in your head, but this problem, it's
 still not trivial to do in your head.
 So let's make the substitution.
 Let's substitute this right here with a u.
 So let's do that.
 So if you define u, and it doesn't have to be u, it's
 just, that's the convention, it's called usubstitution, it
 could have been ssubstitution for all we care.
 Let's say u is equal to the natural log of x, and then du
 dx, the derivative of u with respect to x, of course
 is equal to 1/x.
 Or, just the differential du, if we just multiply both sides
 by dx is equal to 1 over x dx.
 So let's make our substitution.
 This is our integral.
 So this will be equal to the indefinite integral, or the
 antiderivative, of 2 to the now u, so 2 to the u,
 times 1 over x dx.
 Now what is 1 over x dx?
 That's just du.
 So this term times that term is just our du.
 Let me do it in a different color.
 1 over x times dx is just equal to du.
 That's just equal to that thing, right there.
 Now, this still doesn't look like an easy integral, although
 it's gotten simplified a good bit.
 And to solve, you know, whenever I see the variable
 that I'm integrating against in the exponent, you know,
 we don't have any easy exponent rules here.
 The only thing that I'm familiar with, where I have my
 x or my variable that I'm integrating against in
 my exponent, is the case of e to the x.
 We know that the integral of e to the x, dx, is equal
 to e to the x plus c.
 So if I could somehow turn this into some variation of e to the
 x, maybe, or e to the u, maybe I can make this integral a
 little bit more tractable.
 So let's see.
 How can we redefine this right here?
 Well, 2, 2 is equal to what?
 2 is the same thing as e to the natural log of 2, right?
 The natural log of 2 is the power you have to
 raise e to to get 2.
 So if you raise e to that power, you're, of
 course going to get 2.
 This is actually the definition of really, the natural log.
 You raise e to the natural log of 2, you're going to get 2.
 So let's rewrite this, using this I guess we could call
 this this rewrite or I don't want to call it
 quite a substitution.
 It's just a different way of writing the number 2.
 So this will be equal to, instead of writing the number
 2, I could write e to the natural log of 2.
 And all of that to the u du.
 And now what is this equal to?
 Well, if I take something to an exponent, and then to another
 exponent, this is the same thing as taking my base to the
 product of those exponents.
 So this is equal to, let me switch colors, this is equal
 to the integral of e, to the u, e to the, let
 me write it this way.
 e to the natural log of 2 times u.
 I'm just multiplying these two exponents.
 I raise something to something, then raise it again, we know
 from our exponent rules, it's just a product of
 those two exponents.
 du.
 Now, this is just a constant factor, right here.
 This could be, you know, this could just be some number.
 We could use a calculator to figure out what this is.
 We could set this equal to a.
 But we know in general that the integral, this is pretty
 straightforward, we've now put it in this form.
 The antiderivative of e to the au, du, is just
 1 over a e to the au.
 This comes from this definition up here, and of course plus
 c, and the chain rule.
 If we take the derivative of this, we take the derivative
 of the inside, which is just going to be a.
 We multiply that times the one over a, it cancels
 out, and we're just left with e to the au.
 So this definitely works out.
 So the antiderivative of this thing right here is going to be
 equal to 1 over our a, it's going to be 1 over our constant
 term, 1 over the natural log of 2 times our whole
 expression, e e.
 And I'm going to do something.
 This is just some number times u, so I can write it as
 u times some number.
 And I'm just doing that to put in a form that might help us
 simplify it a little bit.
 So it's e to the u times the natural log of 2, right?
 All I did, is I swapped this order.
 I could have written this as e to the natural
 log of 2 times u.
 If this an a, a times u is the same thing as u times a.
 Plus c.
 So this is our answer, but we have to kind of reverse
 substitute before we can feel satisfied that we've taken
 the antiderivative with respect to x.
 But before I do that, let's see if I can simplify
 this a little bit.
 What is, if I have, just from our natural log properties,
 or logarithms, a times the natural log of b.
 We know this is the same thing as the natural
 log of b to the a.
 Let me draw a line here.
 Right?
 That this becomes the exponent on whatever we're taking
 the natural log of.
 So u, let me write this here, u times the natural log of
 2, is the same thing as the natural log of 2 to the u.
 So we can rewrite our antiderivative as being equal
 to 1 over the natural log of 2, that's just that part here,
 times e to the, this can be rewritten based on this
 logarithm property, as the natural log of 2 to the u, and
 of course we still have our plus c there.
 Now, what is e raised to the natural log of 2 to the u?
 The natural log of 2 to the u is the power that you have to
 raise e to to get to 2 to the u, right?
 By definition!
 So if we raise e to that power, what are we going to get?
 We're going to get 2 to the u.
 So this is going to be equal to 1 over the natural log of 2.
 This simplifies to just 2 to the u.
 I drew it up here.
 The natural log a I could just write in general terms, let
 me do it up here, and maybe I'm beating a dead horse.
 But I can in general write any number a as being equal to
 e to the natural log of a.
 This is the exponent you have to raise e to to get a.
 If you raise e to that, you're going to get a.
 So e to the natural log of 2 to the u, that's just 2 to the u.
 And then I have my plus c, of course.
 And now we can reverse substitute.
 What did we set u equal to?
 We defined u, up here, as equal to the natural log of x.
 So let's just reverse substitute right here.
 And so the answer to our original equation, your answer
 to, let me write it here, because it's satisfying when
 you see it, to this kind of fairly convolutedlooking
 antiderivative problem, 2 to the natural log of x over x dx,
 we now find is equal to, we just replaced u with natural
 log of x, because that was our substitution, and 1 over the
 natural log of 2 times 2 to the natural log of x plus c.
 And we're done.
 This isn't in the denominator, the way I wrote it might
 look a little ambiguous.
 And we're done!
 And that was a pretty neat problem, and so thanks
 to Bud for posting that.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

Have something that's not a question about this content? 
This discussion area is not meant for answering homework questions.
At around 4:50 onwards Sal shows that the anti derivative of e^au is (1/a)e^au. even though in reverse (taking the derivative) it seems to make sense I can't really understand how he gets the 1/a when taking the anti derivative.
Try making the following substitution:
(by int[] i mean the integral of whats inside the square brackets)
int[e^ax(dx)]
let u = ax,
then du = a(dx)
then dx = du/a
now substitute this back into the original integral:
int[e^ax(dx)] now equals int[e^u(du/a)]
Therefore:
int[e^ax(dx)] = (1/a)int[e^u(du)]
(by int[] i mean the integral of whats inside the square brackets)
int[e^ax(dx)]
let u = ax,
then du = a(dx)
then dx = du/a
now substitute this back into the original integral:
int[e^ax(dx)] now equals int[e^u(du/a)]
Therefore:
int[e^ax(dx)] = (1/a)int[e^u(du)]
Just wanted to add that there is no way around the words "in reverse." You're looking for the function whose derivative is {the function before you}.
In a nutshell, when integrating, you're working backwards.
You don't sound sure about the derivative. Always a good idea to review the derivative, ask it the question you have(like, "what happens to the constants when I differentiate ce^ax ?"), and return to the problem.
In a nutshell, when integrating, you're working backwards.
You don't sound sure about the derivative. Always a good idea to review the derivative, ask it the question you have(like, "what happens to the constants when I differentiate ce^ax ?"), and return to the problem.
Why is this video in the Multivariable Calculus playlist?
Probably irrelevant by now, but it's been moved to the Usubstitution section because the solution is found using Usubstitution.
Could we not also say 2^Lnx = (e^Ln2)^Lnx = e^(Ln2*Lnx) = (e^Lnx)^Ln2 = x^Ln2 ?
Then we have the integral of (x^Ln2)/x, which should just be the integral of x^(Ln2  1)dx, which is just a simple power rule (add one to exponent, divide by this new exponent). Might be easier for some people to see it this way, as there is no need for substitution, just Ln algebra instead, which we ended up needing anyways.
Then we have the integral of (x^Ln2)/x, which should just be the integral of x^(Ln2  1)dx, which is just a simple power rule (add one to exponent, divide by this new exponent). Might be easier for some people to see it this way, as there is no need for substitution, just Ln algebra instead, which we ended up needing anyways.
In fact if you have studied properties of logarithms in detail, this is a general rule that:
x^log (y) = y^log (x) [ the base of the logarithm can be anything. ]
which would reduce the number of steps required to solve the integral.
However, I solved it in a similar manner.
x^log (y) = y^log (x) [ the base of the logarithm can be anything. ]
which would reduce the number of steps required to solve the integral.
However, I solved it in a similar manner.
Yep, this was how I solved it.
There is a much easier way to do this problem, that I came up with.
We know that d/dx (a^x) = a^x lna. Then we can figure out that d/dx (a^x/lna) = a^x because lna is a constant and it can be taken out, then it gets cancelled.
So, antiderivative of a^x dx = a^x/lna !
Then we just have to put u = ln x. We get 2^u which is in the form a^x
We know that d/dx (a^x) = a^x lna. Then we can figure out that d/dx (a^x/lna) = a^x because lna is a constant and it can be taken out, then it gets cancelled.
So, antiderivative of a^x dx = a^x/lna !
Then we just have to put u = ln x. We get 2^u which is in the form a^x
@ 3:16 I didnt get how' 2 is the same thing as e to the natural log of 2 ' ?can someone please explain?
e^x and lnx are opposite functions. It would be akin to multiplying by some number and then dividing by that same number. The operations cancel one another out.
e to the natural log of *elephant* = elephant.
e to the natural log of *elephant* = elephant.
Natural log of x means "e to the what power equals x"? In this problem, e^ln(2)=2 out because this is what you're doing: e to the what power equals 2? Well, I don't know, but I'm going to take e to that power. e to the power that makes it equal to 2 is obviously 2.
On the other hand, taking the natural log of e^2 is also equal to 2. So, you are asking, "e to the what power equals e^2?". e squared is equal to e squared. So, that would also be just 2.
On the other hand, taking the natural log of e^2 is also equal to 2. So, you are asking, "e to the what power equals e^2?". e squared is equal to e squared. So, that would also be just 2.
how to integrate ln(x)^2
you have to use recursive integration by parts(which is product rule in reverse):
⌠ln(x)^2dx = ln(x)•⌠ln(x)dx  ⌠ln(x)/x dx;
⌠ln(x)dx = x•ln(x)  ⌠x/xdx= x•ln(x)  x (+C) //we plug this back in
⌠ln(x)^2dx = ln(x)•(x•ln(x)  x)  ⌠ln(x)/x dx; u=ln(x), u' = 1/x; dx=du/(1/x);
. . . . . . . . =ln(x)•(x•ln(x)  x)  ⌠1/(x/x)du;
. . . . . . . . =ln(x)•(x•ln(x)  x)  u +C
. . . . . . . . =ln(x)•(x•ln(x)  x) ln(x) +C
. . . . . . . . =ln(x)•(x•ln(x)  x  1)+C ; hard, but not as bad as (sec(x))^3.
⌠ln(x)^2dx = ln(x)•⌠ln(x)dx  ⌠ln(x)/x dx;
⌠ln(x)dx = x•ln(x)  ⌠x/xdx= x•ln(x)  x (+C) //we plug this back in
⌠ln(x)^2dx = ln(x)•(x•ln(x)  x)  ⌠ln(x)/x dx; u=ln(x), u' = 1/x; dx=du/(1/x);
. . . . . . . . =ln(x)•(x•ln(x)  x)  ⌠1/(x/x)du;
. . . . . . . . =ln(x)•(x•ln(x)  x)  u +C
. . . . . . . . =ln(x)•(x•ln(x)  x) ln(x) +C
. . . . . . . . =ln(x)•(x•ln(x)  x  1)+C ; hard, but not as bad as (sec(x))^3.
Hi, couldn't you use the formula Integral a^x dx = (a^x/ln a) + c
Yes thats correct. Sal was just explaining the concept without using that formula.
You could have taken u=2^ln x differentiated and then substituted right?
I'm not sure what you're suggesting, but I suspect it wouldn't work. To make a usubstitution work, the formula has to include both the thing you're choosing as u and the derivative of u (that is, du). Sal is able to do a usubstitution using ln x here because the formula also includes 1/x, the derivative of ln x. We can't do a usubstitution using 2^(ln x) because the formula doesn't contain anything corresponding to the derivative of that expression.
How to integrate x^2 (ln x)^2
This requires integrating by parts two separate times. Here's how to do it.
∫ x² ln²(x) dx
Integrate by parts: ∫ fdg = fg ∫ gdf, where
f = ln²(x), dg = x²dx,
df = [2 ln(x)]/x dx, g = ⅓x³
= ⅓ ∫ 2 x² ln(x) dx+⅓ x³ ln²(x)
Factor out 2 from integrand:
= ⅔ ∫ x² ln(x) dx+⅓ x³ ln²(x)
Integrate ∫x² ln(x), by parts, ∫ f dg = f g ∫ g df, where
f = ln(x), dg = x² dx,
df = 1/x dx, g = ⅓x³
= (2 ∫ x² dx)/9²⁄₉ x³ ln(x)+⅓ x³ ln²(x)
=(²⁄₉ ∫ x² dx) ²⁄₉ x³ ln(x)+⅓ x³ ln²(x)
The integral of x² is ⅓x³
= (2 x³)[¹⁄₂₇] ²⁄₉ x³ ln(x)+⅓ x³ ln²(x) + C
You could probably leave it there or you can simplify by factoring out ¹⁄₂₇ x³ :
= [¹⁄₂₇] x³ { 2  6ln(x) + 9ln²(x)} + C
∫ x² ln²(x) dx
Integrate by parts: ∫ fdg = fg ∫ gdf, where
f = ln²(x), dg = x²dx,
df = [2 ln(x)]/x dx, g = ⅓x³
= ⅓ ∫ 2 x² ln(x) dx+⅓ x³ ln²(x)
Factor out 2 from integrand:
= ⅔ ∫ x² ln(x) dx+⅓ x³ ln²(x)
Integrate ∫x² ln(x), by parts, ∫ f dg = f g ∫ g df, where
f = ln(x), dg = x² dx,
df = 1/x dx, g = ⅓x³
= (2 ∫ x² dx)/9²⁄₉ x³ ln(x)+⅓ x³ ln²(x)
=(²⁄₉ ∫ x² dx) ²⁄₉ x³ ln(x)+⅓ x³ ln²(x)
The integral of x² is ⅓x³
= (2 x³)[¹⁄₂₇] ²⁄₉ x³ ln(x)+⅓ x³ ln²(x) + C
You could probably leave it there or you can simplify by factoring out ¹⁄₂₇ x³ :
= [¹⁄₂₇] x³ { 2  6ln(x) + 9ln²(x)} + C
Couldn't the answer also be ((2^lnx)/(ln2)) + C ?
Yes, that's just a different way of writing the solution Sal demonstrates in this video.
I don't get what he did from 3:36 onwards. Could someone please explain?
When I use int[] I mean to say the indefinite integral of whatever is in the [] brackets.
At 3:36 we have int[2^u du] which looks relatively simplified but we aren't too sure as to what that is, but we are familiar with integrals involving the powers of e.
So a nice way to rewrite 2 is to raise e to the ln(2) power. Because ln(2) gives us the value "what we have to raise e by to get 2", so if we raise e to that power, we're logically going to get 2. So e^ln(2) = 2. Therefore we can rewrite int[2^u du] as:
int[(e^ln(2))^ u du]. By exponent properties, it is equivalent to: [int e^(ln(2) * u) du]
We know that int[e^au du] = 1/a * e^au (There is proof of this in one of the top comments). So, the current integral that we have can be rewritten as:
1/ln(2) * (e^(ln(2) * u)). As was established earlier, e^ln(2) = 2. so e^ln(2)*u = (e^ln(2))^u = 2^u.
So in the end we have 1/ln(2) * 2^u. Now we plug back what u is and we get
1/ln(2) * 2^(ln(x))
At 3:36 we have int[2^u du] which looks relatively simplified but we aren't too sure as to what that is, but we are familiar with integrals involving the powers of e.
So a nice way to rewrite 2 is to raise e to the ln(2) power. Because ln(2) gives us the value "what we have to raise e by to get 2", so if we raise e to that power, we're logically going to get 2. So e^ln(2) = 2. Therefore we can rewrite int[2^u du] as:
int[(e^ln(2))^ u du]. By exponent properties, it is equivalent to: [int e^(ln(2) * u) du]
We know that int[e^au du] = 1/a * e^au (There is proof of this in one of the top comments). So, the current integral that we have can be rewritten as:
1/ln(2) * (e^(ln(2) * u)). As was established earlier, e^ln(2) = 2. so e^ln(2)*u = (e^ln(2))^u = 2^u.
So in the end we have 1/ln(2) * 2^u. Now we plug back what u is and we get
1/ln(2) * 2^(ln(x))
has this type of question any general form ? an easy and convenient way for solving it down in multiple choice questions ?
Why is the problem expressed as (2^ln x)/x, when this is equivalent to (x^ln 2)/x = x^(ln 2  1), which is simpler?
My change from (2^ln x)/x to x^(ln(2)  1) is mostly just the power rules:
2^(ln x) = (e^ln 2)^(ln x) = e^((ln 2) * (ln x)) = (e^ln x)^(ln 2) = x^(ln 2)
(x^(ln 2))/x = (x^(ln 2)) * x^(1) = x^(ln(2)  1)
WolframAlpha verifies that this is true: http://www.wolframalpha.com/input/?i=is+%282%5E%28ln+x%29%29%2Fx+%3D+x%5E%28ln%282%29++1%29
2^(ln x) = (e^ln 2)^(ln x) = e^((ln 2) * (ln x)) = (e^ln x)^(ln 2) = x^(ln 2)
(x^(ln 2))/x = (x^(ln 2)) * x^(1) = x^(ln(2)  1)
WolframAlpha verifies that this is true: http://www.wolframalpha.com/input/?i=is+%282%5E%28ln+x%29%29%2Fx+%3D+x%5E%28ln%282%29++1%29
What exact rule are you using to say that (x^ln 2)/x = x^(ln 2  1)? If this was set up in any form of addition or multiplication where the log rules apply, I could see some of those things swapping around. However, because this is something to a log over something that does not involve a log, I do not see how you are saying they are equal.
Sal, when you get the integral of 2^u dx, can't you just use the reverse power rule and get (2^(u+1))/(u+1)?
No, the reverse power rule is for when the variable is the base, not the exponent.
when Sal gets to 1/(ln 2) * (e^(u ln 2)) + C, can't he just substitute lnx for u, getting
1/(ln 2) * (e^(ln x * ln 2 )) + C, or 1/(ln 2) * ((e^(ln x))^(ln2) + C. Then won't he have 1/ln2 * x^(ln 2) + C?
1/(ln 2) * (e^(ln x * ln 2 )) + C, or 1/(ln 2) * ((e^(ln x))^(ln2) + C. Then won't he have 1/ln2 * x^(ln 2) + C?
when he has `S2^udu`, why can't he just use the reverse product rule??
Can someone please explain me the step Sal takes at 6:19, where he takes the integral? I don't really understand how he got there.
integration of e to the power root x
This problem takes a somewhat strange substitution.
Let u = sqrt(x). Before substituting, notice that u^2 = x.
Thus, 2u du = dx.
Now when you substitute, you get:
integral(e^sqrt(x)) dx = integral(e^u*2u du) = 2integral(u*e^u du).
This integral still requires integration by parts. Hopefully you can proceed from here. If you don't know integration by parts yet, back away from this problem!
Let u = sqrt(x). Before substituting, notice that u^2 = x.
Thus, 2u du = dx.
Now when you substitute, you get:
integral(e^sqrt(x)) dx = integral(e^u*2u du) = 2integral(u*e^u du).
This integral still requires integration by parts. Hopefully you can proceed from here. If you don't know integration by parts yet, back away from this problem!
At 3:20 how do we know that we have to rewrite 2^u? Why do you see that it should be in the format of Intergral(e^2)dx?
Are there any methods to solving this particular question?
at 1900, may i please get someone to explain this problem over video chat?
at 4:45 is Sal writing (e^ln2)^u or (e)^ln2*u? I mean e^ln2 equals 2 and 2 is raised to the power u...so why should it be the latter and not the former?
if you remember your logarithmic rules, you will know that log(x^y) = y*log(x)
The method Sal uses is such a complex way of doing it. Just take u = 2^(lnx). Then dx/x = 1/(uin2), so the integral simplifies to du/ln2 which, once integrated, is equal to u/ln 2 = (2^lnx)/ln2. It's a far easier way of doing it and is simpler because it doesn't have all the e's in it. Using this substitution and method will save you about 5 minutes and a lot of brain power.
i don't understand how only 1/ln2 came down and nothing else at 6.11 in the video..
At 5:00, why does the integral of e^(au) become 1/a * e^(au)? Where did the 1/a come from? Thanks!
The chain rule, I presume would be the answer, since the expression uses a function of a function ( the thing you wrote up). Hope that helps :)
can you make a video on some standard integral with some examples
This is the playlist for definite and indefinite integrals
https://www.khanacademy.org/math/calculus/integralcalculus
https://www.khanacademy.org/math/calculus/integralcalculus
Is there a way to check your answer after you find the answer from the USubs?
You can check your answer to any kind of integration problem by differentiating your answer. The derivative will equal the original integrand if you did everything correctly. This particular result is somewhat tricky to differentiate, but you can do it if you remember that 1/ln2 is just a constant and 2^lnx can be rewritten as e^(ln2lnx).
Color of the constant is constantly changing :)
what would be the integral of a^x^2 dx??
u could have used the standard integral of a^x=a^x/lna
Am I correct in assuming instead of using e^ln2 we could have just multiplied the integral by 1/2 and then multiplied it by e, to make 2 into e, and put an "2*(1/e)" on the outside to distribute back in at the end?
Hi!
I just wanna ask, that why its needed to use the form of the e^ln2? Because already there is a form saying :int a^x dx= (a^x/ln a)+C
I just wanna ask, that why its needed to use the form of the e^ln2? Because already there is a form saying :int a^x dx= (a^x/ln a)+C
can someone please explain why what the logic is in rewriting 2^u to e^ln 2?
Uhhh... I think you missed part. He did not write 2^u as e^ln 2. He wrote it as (e^ln 2)^u, which lets use exponent and log rules to rewite it into a something that has an easy to evaluate item in it: e. Then, once the integral of e is done, he simplifies it back to gets the 2^u/(ln 2).
what does that big line stand for
The yellow one he draws by alnb = lnb^a? That's just showing that the two things are entirely separate, do not mix them.
do you mean the integral sign? It means the counter derivative  what you need to derive in order to get the thing inside the line
hey i got another result for the integral:
(x^(ln 2))/(ln 2) + c
and when i plot the function in wolfram alpha it shows the exact same function as when I plot
sal's result!! Did anyone else got the same result?
(x^(ln 2))/(ln 2) + c
and when i plot the function in wolfram alpha it shows the exact same function as when I plot
sal's result!! Did anyone else got the same result?
can i get someone to explain this in video chat please?
Couldn't you just notice that d/dx 2^ln x= (ln(2)*2^ln(x))/x and correct for the ln(2) to get the antiderivative of (2^ln(x))/x is (2^ln x)/ln(2)?(and to stave off some pointless comments, ln(2)is a constant.)
what is the antiderivetive of lnsec(x)dx?
nevermind. I should watch the video before I ask questions.
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