Main content

### Course: AP®︎/College Calculus BC > Unit 6

Lesson 13: Using integration by parts- Integration by parts intro
- Integration by parts: ∫x⋅cos(x)dx
- Integration by parts: ∫ln(x)dx
- Integration by parts: ∫x²⋅𝑒ˣdx
- Integration by parts: ∫𝑒ˣ⋅cos(x)dx
- Integration by parts
- Integration by parts: definite integrals
- Integration by parts: definite integrals
- Integration by parts challenge
- Integration by parts review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Integration by parts: ∫ln(x)dx

This video shows how to find the antiderivative of the natural log of x using integration by parts. We rewrite the integral as ln(x) times 1dx, then choose f(x) = ln(x) and g'(x) = 1. The antiderivative is xln(x) - x + C. Created by Sal Khan.

## Want to join the conversation?

- f(x) = 1/ln(x),

is there a vertical asymptote at x=0 ?

I think that there is...is there?

Also, is the point x=1 a global minima?

I think not because ln(1) = 0 and therefore f(x) is not define there so it can't be minima.

What do you think?

Thanks.(8 votes)- I don't think there is a vertical asymptote at x=0 because lim(x-->0+) f(x) = 0(3 votes)

- When you differentiate the end result, don't you get ln(x)-1 rather than ln(x)?(12 votes)
- The calculation follows the chain rule : d/dx (x ln x ) = 1 * ln x + x * 1/x = ln x + 1

So, in d/dx (x ln x - x) you have to add d/dx (-x) = -1

Together : = ln x + 1 - 1 = ln x(4 votes)

- Can xlnx be written as ln x^2 ?

If not then why ?(1 vote)- If you are trying to use properties of logarithms, you would bring the "x" from the front into the logarithm as an exponent, resulting in:

ln (x^x)(37 votes)

- At2:58sal say the word integrand. What is the difference between an integral and an integrand?(10 votes)
- An integral is the whole operator: ∫ f(x) dx

An integrand is just the function you are integrating. So for ∫ 3x^2 dx, the integrand is 3x^2.(15 votes)

- At2:20he integrated g'(x)=1 to get g(x)=x, but shouldn't the integral of g'(x)=1 be g(x)=x+c?(8 votes)
- That is correct if that is where you were going to end your problem, but since there will be further integrals down the road, you can just add a +C at the end of the problem to encompass all the +C you would have had to put in.(5 votes)

- why do you consider 1 as a function in this case and not in other cases?(6 votes)
- You are going to see more and more of this if you continue in math, that is, the creative use of the rules and properties of numbers and processes. In this case, treating the 1 as the result of differentiating some function g(x)=x, made it possible the use of integration by parts to solve the problem. Use whatever works to solve problems. Get creative. But stay within the rules. For me, this is the most fun part of math where you can unleash your creativity! At its best, it is the playground of new ideas, at its worst, it is where you hone your intuition by learning what works and what doesn't - and that isn't bad at all!(8 votes)

- Hi, just doing some revision for this, I always thought that the integral of Ln(x) was always 1/x?(2 votes)
- No, the
*derivative*of ln(x) is 1/x. As Sal points out here, ∫ lnx dx is

xlnx-x+c(9 votes)

- How do I know which part of the function is f(x) and which is g'(x)? I always end up trying both possibilities.(3 votes)
- You need to develop an intuition for which function will simplify with either taking the derivative or the anti-derivative. It's a matter of practice.(5 votes)

- I am 65 years old, and re-learning calculus on my own, using an old textbook.

When I first come upon integrating ln x, the book has not yet mentioned integration by parts. How would you integrate ln x without using integration by parts?(5 votes)- Awesome! I'm in my mid 50's.

Your book probably just provided the cookbook result. I'm not aware of any other method to compute the integral other than IBP.(2 votes)

- If the derivitive of a function is written as the prime of that function (i.e. f'(x)) is there such notation for the antiderivitive of a function?(2 votes)
- The antiderivative is the same as an integral. You can explore this here : https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/indefinite_integrals/v/antiderivatives-and-indefinite-integrals

:)(5 votes)

## Video transcript

The goal of this video
is to try to figure out the antiderivative of
the natural log of x. And it's not completely obvious
how to approach this at first, even if I were to tell you
to use integration by parts, you'll say,
integration by parts, you're looking for the
antiderivative of something that can be expressed as the
product of two functions. It looks like I only
have one function right over here, the
natural log of x. But it might become a
little bit more obvious if I were to rewrite
this as the integral of the natural log
of x times 1dx. Now, you do have the
product of two functions. One is a function,
a function of x. It's not actually dependent on
x, it's always going to be 1, but you could have f
of x is equal to 1. And now it might
become a little bit more obvious to use
integration by parts. Integration by parts
tells us that if we have an integral that can be
viewed as the product of one function, and the derivative
of another function, and this is really just
the reverse product rule, and we've shown that
multiple times already. This is going to be equal to
the product of both functions, f of x times g of x minus
the antiderivative of, instead of having f
and g prime, you're going to have f prime and g. So f prime of x times g of x dx. And we've seen this
multiple times. So when you figure
out what should be f and what should
be g, for f you want to figure out
something that it's easy to take the derivative
of and it simplifies things, possibly if you're taking
the derivative of it. And for g prime of x, you
want to find something where it's easy to take
the antiderivative of it. So good candidate for f
of x is natural log of x. If you were to take
the derivative of it, it's 1 over x. Let me write this down. So let's say that f of x is
equal to the natural log of x. Then f prime of x is
equal to 1 over x. And let's set g prime
of x is equal to 1. So g prime of x is equal to 1. That means that g of
x could be equal to x. And so let's go back
right over here. So this is going to be equal
to f of x times g of x. Well, f of x times g of
x is x natural log of x. So g of x is x, and f of
x is the natural log of x, I just like writing
the x in front of the natural log of
x to avoid ambiguity. So this is x natural log of x
minus the antiderivative of f prime of x, which
is 1 over x times g of x, which is x, which is xdx. Well, what's this
going to be equal to? Well, what we have inside
the integrand, this is just 1 over x times x, which
is just equal to 1. So this simplifies quite nicely. This is going to end up
equaling x natural log of x minus the antiderivative
of, just dx, or the antiderivative of
1dx, or the integral of 1dx, or the antiderivative
of 1 is just minus x. And this is just an
antiderivative of this. If we want to write the entire
class of antiderivatives we just have to add a plus
c here, and we are done. We figured out
the antiderivative of the natural log of x. I encourage you to take
the derivative of this. For this part, you're going
to use the product rule and verify that you do indeed
get natural log of x when you take the derivative of this.