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.