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Studying for a test? Prepare with these 12 lessons on Antiderivatives and the fundamental theorem of calculus.

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# Integration by parts: ∫ln(x)dx

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.