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:

1. Integration by parts tells us that ∫ udv = uv - ∫ vdu
2. Let’s set u = ln(x) and dv = dx
3. du = (1/x)dx and v = x
4. ∫ ln(x)dx = uv - ∫ vdu = xln(x) - ∫ x(1/x)dx
5. xln(x) - ∫ dx
6. xln(x) - x + C
7. 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.