Integralen av $$$\frac{3 x^{2} - 209 x}{x^{2}}$$$

Bestäm $$$\int \frac{3 x^{2} - 209 x}{x^{2}}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{3 x^{2} - 209 x}{x^{2}} d x}}} = {\color{red}{\int{\left(3 - \frac{209}{x}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(3 - \frac{209}{x}\right)d x}}} = {\color{red}{\left(\int{3 d x} - \int{\frac{209}{x} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=3$$$:

$$- \int{\frac{209}{x} d x} + {\color{red}{\int{3 d x}}} = - \int{\frac{209}{x} d x} + {\color{red}{\left(3 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=209$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$3 x - {\color{red}{\int{\frac{209}{x} d x}}} = 3 x - {\color{red}{\left(209 \int{\frac{1}{x} d x}\right)}}$$

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

$$3 x - 209 {\color{red}{\int{\frac{1}{x} d x}}} = 3 x - 209 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Alltså,

$$\int{\frac{3 x^{2} - 209 x}{x^{2}} d x} = 3 x - 209 \ln{\left(\left|{x}\right| \right)}$$

Lägg till integrationskonstanten:

$$\int{\frac{3 x^{2} - 209 x}{x^{2}} d x} = 3 x - 209 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{3 x^{2} - 209 x}{x^{2}}\, dx = \left(3 x - 209 \ln\left(\left|{x}\right|\right)\right) + C$$$A