Integral von $$$\frac{3}{x^{2} + 2}$$$

Bestimme $$$\int \frac{3}{x^{2} + 2}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=3$$$ und $$$f{\left(x \right)} = \frac{1}{x^{2} + 2}$$$ an:

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

Sei $$$u=\frac{\sqrt{2}}{2} x$$$.

Dann $$$du=\left(\frac{\sqrt{2}}{2} x\right)^{\prime }dx = \frac{\sqrt{2}}{2} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$dx = \sqrt{2} du$$$.

Das Integral lässt sich umschreiben als

$$3 {\color{red}{\int{\frac{1}{x^{2} + 2} d x}}} = 3 {\color{red}{\int{\frac{\sqrt{2}}{2 \left(u^{2} + 1\right)} d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{\sqrt{2}}{2}$$$ und $$$f{\left(u \right)} = \frac{1}{u^{2} + 1}$$$ an:

$$3 {\color{red}{\int{\frac{\sqrt{2}}{2 \left(u^{2} + 1\right)} d u}}} = 3 {\color{red}{\left(\frac{\sqrt{2} \int{\frac{1}{u^{2} + 1} d u}}{2}\right)}}$$

Das Integral von $$$\frac{1}{u^{2} + 1}$$$ ist $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

$$\frac{3 \sqrt{2} {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{2} = \frac{3 \sqrt{2} {\color{red}{\operatorname{atan}{\left(u \right)}}}}{2}$$

Zur Erinnerung: $$$u=\frac{\sqrt{2}}{2} x$$$:

$$\frac{3 \sqrt{2} \operatorname{atan}{\left({\color{red}{u}} \right)}}{2} = \frac{3 \sqrt{2} \operatorname{atan}{\left({\color{red}{\frac{\sqrt{2}}{2} x}} \right)}}{2}$$

Daher,

$$\int{\frac{3}{x^{2} + 2} d x} = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{3}{x^{2} + 2} d x} = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}+C$$

$$$\int \frac{3}{x^{2} + 2}\, dx = \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + C$$$A