Integral of ln x

Q: What is the integral of ln x?

\int \ln\left(x\right)\,dx = x \ln\left(x\right) - x + C.

Q: How can I check the answer?

Differentiate it: the derivative of x \ln\left(x\right) - x is \ln\left(x\right). CalcViz checks every antiderivative this way before showing it.

\int \ln\left(x\right)\,dx = x \ln\left(x\right) - x + C

ln x has no simple "reverse rule". The trick is integration by parts with u = ln x and dv = dx, so the logarithm gets differentiated into 1/x and disappears.

Step by step

Integration by parts (known result)
With u = ln x, dv = dx: ∫ln x dx = x·ln x − x.
$\int \ln\left(x\right)\,dx = x \ln\left(x\right) - x$

f(x) = ln x
F(x) + C

Constant C = 0.0

Every choice of C gives a valid antiderivative, since shifting up or down doesn’t change the slope.

Open in the integral calculator

Questions students ask

What is the integral of ln x?

$\int \ln\left(x\right)\,dx = x \ln\left(x\right) - x + C$ .

How do you integrate ln x?

ln x has no simple "reverse rule". The trick is integration by parts with u = ln x and dv = dx, so the logarithm gets differentiated into 1/x and disappears.

How can I check the answer?

Differentiate it: the derivative of $x \ln\left(x\right) - x$ is $\ln\left(x\right)$ . CalcViz checks every antiderivative this way before showing it.