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

Find $$$\int \frac{\ln\left(x\right)}{x^{2}}\, dx$$$.

For the integral $$$\int{\frac{\ln{\left(x \right)}}{x^{2}} d x}$$$, 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 $$$\operatorname{dv}=\frac{dx}{x^{2}}$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\frac{1}{x^{2}} d x}=- \frac{1}{x}$$$ (steps can be seen »).

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-2$$$:

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

$$$\int \frac{\ln\left(x\right)}{x^{2}}\, dx = \frac{- \ln\left(x\right) - 1}{x} + C$$$A