Integraali $$$- 3 x_{2} + \frac{1}{x}$$$:stä muuttujan $$$x$$$ suhteen

Määritä $$$\int \left(- 3 x_{2} + \frac{1}{x}\right)\, dx$$$.

Integroi termi kerrallaan:

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

Funktion $$$\frac{1}{x}$$$ integraali on $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Sovella vakiosääntöä $$$\int c\, dx = c x$$$ käyttäen $$$c=3 x_{2}$$$:

$$\ln{\left(\left|{x}\right| \right)} - {\color{red}{\int{3 x_{2} d x}}} = \ln{\left(\left|{x}\right| \right)} - {\color{red}{\left(3 x x_{2}\right)}}$$

Näin ollen,

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

Lisää integrointivakio:

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

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