Integralen av $$$- \frac{x^{2}}{2} + \frac{1}{x}$$$

Bestäm $$$\int \left(- \frac{x^{2}}{2} + \frac{1}{x}\right)\, dx$$$.

Integrera termvis:

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

Integralen av $$$\frac{1}{x}$$$ är $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

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

Alltså,

$$\int{\left(- \frac{x^{2}}{2} + \frac{1}{x}\right)d x} = - \frac{x^{3}}{6} + \ln{\left(\left|{x}\right| \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{x^{2}}{2} + \frac{1}{x}\right)d x} = - \frac{x^{3}}{6} + \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(- \frac{x^{2}}{2} + \frac{1}{x}\right)\, dx = \left(- \frac{x^{3}}{6} + \ln\left(\left|{x}\right|\right)\right) + C$$$A