Question

Solve the integral: $\int x\ln \left(x^2\right)dx$

Short answer

The required answer is$x^2\ln \left(x\right)-\frac{x^2}{2}+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int x\ln \left(x^2\right)dx$.

Step 2. Solve the integration by parts . 

We have,

$$\begin{gathered} u=\ln \left(x\right) \\ du=\frac{dx}{x} \end{gathered}$$

and

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

The formula of integration by parts is $\int udv=uv-\int vdu$
$$\begin{gathered} \int x\ln \left(x^2\right)dx \\ =2\left(\ln \left(x\right)\frac{x^2}{2}-\int \frac{x^2}{2}\times \frac{1}{x}dx\right) \\ =2\left(\ln \left(x\right)\frac{x^2}{2}-\int \frac{x}{2}dx\right) \\ =x^2\ln \left(x\right)-\frac{x^2}{2}+c \end{gathered}$$