Integraal van $$$\frac{2}{x^{2} + 1}$$$

Bepaal $$$\int \frac{2}{x^{2} + 1}\, dx$$$.

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

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

De integraal van $$$\frac{1}{x^{2} + 1}$$$ is $$$\int{\frac{1}{x^{2} + 1} d x} = \operatorname{atan}{\left(x \right)}$$$:

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

Dus,

$$\int{\frac{2}{x^{2} + 1} d x} = 2 \operatorname{atan}{\left(x \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{2}{x^{2} + 1} d x} = 2 \operatorname{atan}{\left(x \right)}+C$$

$$$\int \frac{2}{x^{2} + 1}\, dx = 2 \operatorname{atan}{\left(x \right)} + C$$$A