Integralen av $$$5 x - \frac{5}{x}$$$

Bestäm $$$\int \left(5 x - \frac{5}{x}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(5 x - \frac{5}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{5}{x} d x} + \int{5 x d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\int{5 x d x} - {\color{red}{\int{\frac{5}{x} d x}}} = \int{5 x d x} - {\color{red}{\left(5 \int{\frac{1}{x} d x}\right)}}$$

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

$$\int{5 x d x} - 5 {\color{red}{\int{\frac{1}{x} d x}}} = \int{5 x d x} - 5 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = x$$$:

$$- 5 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{5 x d x}}} = - 5 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(5 \int{x d x}\right)}}$$

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

$$- 5 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\int{x d x}}}=- 5 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 5 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(5 x - \frac{5}{x}\right)d x} = \frac{5 x^{2}}{2} - 5 \ln{\left(\left|{x}\right| \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(5 x - \frac{5}{x}\right)d x} = \frac{5 x^{2}}{2} - 5 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(5 x - \frac{5}{x}\right)\, dx = \left(\frac{5 x^{2}}{2} - 5 \ln\left(\left|{x}\right|\right)\right) + C$$$A