Integral of $$$\frac{3 \ln\left(x\right)}{2 x^{2}}$$$

Find $$$\int \frac{3 \ln\left(x\right)}{2 x^{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{3}{2}$$$ and $$$f{\left(x \right)} = \frac{\ln{\left(x \right)}}{x^{2}}$$$:

$${\color{red}{\int{\frac{3 \ln{\left(x \right)}}{2 x^{2}} d x}}} = {\color{red}{\left(\frac{3 \int{\frac{\ln{\left(x \right)}}{x^{2}} d x}}{2}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{2}} d x}}}}{2}=\frac{3 {\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{x}\right)-\int{\left(- \frac{1}{x}\right) \cdot \frac{1}{x} d x}\right)}}}{2}=\frac{3 {\color{red}{\left(- \int{\left(- \frac{1}{x^{2}}\right)d x} - \frac{\ln{\left(x \right)}}{x}\right)}}}{2}$$

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

$$- \frac{3 {\color{red}{\int{\left(- \frac{1}{x^{2}}\right)d x}}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x} = - \frac{3 {\color{red}{\left(- \int{\frac{1}{x^{2}} d x}\right)}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}$$

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

$$\frac{3 {\color{red}{\int{\frac{1}{x^{2}} d x}}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}=\frac{3 {\color{red}{\int{x^{-2} d x}}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}=\frac{3 {\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}=\frac{3 {\color{red}{\left(- x^{-1}\right)}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}=\frac{3 {\color{red}{\left(- \frac{1}{x}\right)}}}{2} - \frac{3 \ln{\left(x \right)}}{2 x}$$

Therefore,

$$\int{\frac{3 \ln{\left(x \right)}}{2 x^{2}} d x} = - \frac{3 \ln{\left(x \right)}}{2 x} - \frac{3}{2 x}$$

Simplify:

$$\int{\frac{3 \ln{\left(x \right)}}{2 x^{2}} d x} = \frac{3 \left(- \ln{\left(x \right)} - 1\right)}{2 x}$$

Add the constant of integration:

$$\int{\frac{3 \ln{\left(x \right)}}{2 x^{2}} d x} = \frac{3 \left(- \ln{\left(x \right)} - 1\right)}{2 x}+C$$

$$$\int \frac{3 \ln\left(x\right)}{2 x^{2}}\, dx = \frac{3 \left(- \ln\left(x\right) - 1\right)}{2 x} + C$$$A