## Integral of ln(x) Lesson

### How to Solve the Integral of ln(x)

The indefinite integral of ln(x) is given as:

**∫ ln(x)dx = xln(x) – x + C**

The constant of integration *C* is shown because it is the indefinite integral. If taking the definite integral of ln(x), you don't need the *C*.

**There is no integral rule or shortcut that directly gets us to the integral of ln(x).** When finding the definite or indefinite integral of the function f(x) = ln(x), we must use integration by parts. Integrating ln(x) by parts is shown in depth in the section below.

#### Example Problem: Using Integration by Parts to Derive the Integral of ln(x)

The example problem below shows a derivation of the antiderivative of ln(x) in the context of an indefinite integral. However, the steps shown will work for a definite *and* indefinite integral of ln(x). As stated earlier, just leave off the constant of integration *C* if a definite integral is being taken.

Let’s solve ∫ ln(x)dx using integration by parts.

Solution:

- Integration by parts tells us that ∫ udv = uv - ∫ vdu
- Let’s set u = ln(x) and dv = dx
- du = (1/x)dx and v = x
- ∫ ln(x)dx = uv - ∫ vdu = xln(x) - ∫ x(1/x)dx
- xln(x) - ∫ dx
- xln(x) - x + C
- Our final answer is
**xln(x) – x + C.**

This final answer can be memorized as the formula for ∫ ln(x)dx. Keep in mind that it will not work for ln(u) where *u* is any single variable function. The integral we have solved here only applies to the single variable function of the variable itself. **The variable x can be replaced with any other variable, such as y, z, w, θ, etc.**