# Integral of ln x

How to find the integral of ln x: formula, 1 example, and its solution.

## Formula

∫ ln x dx = x ln x - x + C

## Example∫ ln x dx

Solution

Solution (Detail)

ln x = (ln x)⋅1

So solve this

by using integral by parts.

The order of uv' is

[u]

logarithmic (ln x)

polynomial (1)

trigonometric

exponential

[v'].

So set

u = ln x and v' = 1.

Write u = ln x.

Differentiate both sides.

Then u' = 1/x.

Derivative of ln x

Write v' = 1

next to u' = 1/x.

Integrate both sides.

Then v = x.

Integral of a Polynomial

Write this above v' = 1.

u = ln x, v = x

u' = 1/x

Then the given integral is equal to,

uv, (ln x)⋅x

minus

integral, u'v, (1/x)⋅x dx.

(ln x)⋅x = x ln x

(1/x)⋅x = 1

-∫ 1 dx = -x + C

So the integral of ln x is

x ln x - x + C.