Integraal van $$$20 x^{2} - \frac{2}{x}$$$

Bepaal $$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(20 x^{2} - \frac{2}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{2}{x} d x} + \int{20 x^{2} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\int{20 x^{2} d x} - {\color{red}{\int{\frac{2}{x} d x}}} = \int{20 x^{2} d x} - {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

De integraal van $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$\int{20 x^{2} d x} - 2 {\color{red}{\int{\frac{1}{x} d x}}} = \int{20 x^{2} d x} - 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=20$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$$- 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{20 x^{2} d x}}} = - 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(20 \int{x^{2} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\int{x^{2} d x}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Dus,

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx = \left(\frac{20 x^{3}}{3} - 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A