# Integration of lnx

In this tutorial we shall explain integration of the natural logarithmic function $$\ln x$$. It is an important integral function, but there is no direct method to find it. We shall find the integration of $$lnx$$ by using the integration by parts method.

The integration of $$lnx$$ is of the form

\[I = \int {\ln xdx} \]

When using integration by parts it must have at least two functions, however here there is only one function: $$\ln x$$. So consider the second function as $$1$$. Now the integration becomes

\[I = \int {\ln x \cdot 1dx} \,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

The first function is $$\ln x$$ and the second function is $$1$$

Using the formula for integration by parts, we have

\[\int {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int {g\left( x \right)dx – \int {\left[ {\frac{d}{{dx}}f\left( x \right)\int {g\left( x \right)dx} } \right]dx} } } \]

Using the formula above, equation (i) becomes

\[\begin{gathered} I = \ln x\int {1dx – \int {\left[ {\frac{d}{{dx}}\ln x\int {1dx} } \right]dx} } \\ \Rightarrow I = x\ln x – \int {\left[ {\frac{1}{x}x} \right]dx} \\ \Rightarrow I = x\ln x – \int {1dx} \\ \Rightarrow I = x\ln x – x + c \\ \Rightarrow \int {\ln xdx} = x\ln x – x + c \\ \end{gathered} \]

Now in further study of integration we can use this integration of $$lnx$$ as a formula.