Integraal van $$$\frac{1}{x^{2} + 4}$$$

Bepaal $$$\int \frac{1}{x^{2} + 4}\, dx$$$.

Zij $$$u=\frac{x}{2}$$$.

Dan $$$du=\left(\frac{x}{2}\right)^{\prime }dx = \frac{dx}{2}$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = 2 du$$$.

Dus,

$${\color{red}{\int{\frac{1}{x^{2} + 4} d x}}} = {\color{red}{\int{\frac{1}{2 \left(u^{2} + 1\right)} d u}}}$$

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

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

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

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

We herinneren eraan dat $$$u=\frac{x}{2}$$$:

$$\frac{\operatorname{atan}{\left({\color{red}{u}} \right)}}{2} = \frac{\operatorname{atan}{\left({\color{red}{\left(\frac{x}{2}\right)}} \right)}}{2}$$

Dus,

$$\int{\frac{1}{x^{2} + 4} d x} = \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{x^{2} + 4} d x} = \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}+C$$

$$$\int \frac{1}{x^{2} + 4}\, dx = \frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2} + C$$$A