Integration of x ln x

In this tutorial we shall find integral of x ln x, and solve this problem with the help of integration by parts methods

The integral of x ln x is of the form

I = \int {x\ln xdx}

Here first function is \ln x and second function will be x

I = \int {\ln x \cdot xdx} \,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Using 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} } }

Equation (i) becomes using above formula, we have

\begin{gathered} I = \ln x\int {xdx - \int {\left[ {\frac{d}{{dx}}\ln x\left( {\int {xdx} } \right)} \right]} dx} \\ \Rightarrow I = \ln x\frac{{{x^2}}}{2} - \int {\left[ {\frac{1}{x}\frac{{{x^2}}}{2}} \right]} dx \\ \Rightarrow I = \frac{{{x^2}}}{2}\ln x - \frac{1}{2}\int x dx \\ \Rightarrow I = \frac{{{x^2}}}{2}\ln x - \frac{1}{2}\frac{{{x^2}}}{2} + c \\ \Rightarrow I = \frac{{{x^2}}}{2}\ln x - \frac{1}{4}{x^2} + c \\ \end{gathered}