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.