Integral of $$$\ln\left(x^{2}\right)$$$

Find $$$\int 2 \ln\left(x\right)\, dx$$$.

The input is rewritten: $$$\int{\ln{\left(x^{2} \right)} d x}=\int{2 \ln{\left(x \right)} d x}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

$$\color{red}{\int{2 \ln{\left(x \right)} d x}} = \color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}$$

For the integral $$$\int{\ln{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen here) and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen here).

The integral becomes

$$2 \color{red}{\int{\ln{\left(x \right)} d x}}=2 \color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}=2 \color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$2 x \ln{\left(x \right)} - 2 \color{red}{\int{1 d x}} = 2 x \ln{\left(x \right)} - 2 \color{red}{x}$$

Therefore,

$$\int{2 \ln{\left(x \right)} d x} = 2 x \ln{\left(\left|{x}\right| \right)} - 2 x$$

Simplify:

$$\int{2 \ln{\left(x \right)} d x} = 2 x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)$$

Add the constant of integration:

$$\int{2 \ln{\left(x \right)} d x} = 2 x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)+C$$

Answer: $$$\int{2 \ln{\left(x \right)} d x}=2 x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)+C$$$