Question

Solve the integral:$\int x\ln \left(x\right)dx$

Short answer

The required answer is$\frac{x^2}{2}\ln x-\frac{x^2}{4}+c$.

Step-by-step solution

Step 1. Given information.  

We have given integral is$\int x\ln \left(x\right)dx$.

Step 2. Solve the integration by parts . 

We have,$u=\ln x,dv=xdx$.

$$\begin{gathered} u=\ln x \\ du=\frac{1}{x}dx \end{gathered}$$

and

$$\begin{gathered} dv=xdx \\ v=\int xdx \\ v=\frac{x^2}{2} \end{gathered}$$

The formula of integration by parts is $\int udv=uv-\int vdu$

$$\begin{gathered} \int x\ln \left(x\right)dx \\ =\ln \left(x\right)\frac{x^2}{2}-\int \frac{x^2}{2}\times \frac{1}{x}dx \\ =\frac{x^2}{2}\ln \left(x\right)-\int \frac{x}{2}dx \\ =\frac{x^2}{2}\ln \left(x\right)-\frac{x^2}{4}+c \end{gathered}$$