Integraal van $$$\frac{4}{15 x^{2} + 27}$$$

Bepaal $$$\int \frac{4}{15 x^{2} + 27}\, dx$$$.

Vereenvoudig de integraand:

$${\color{red}{\int{\frac{4}{15 x^{2} + 27} d x}}} = {\color{red}{\int{\frac{4}{3 \left(5 x^{2} + 9\right)} d x}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{4}{3}$$$ en $$$f{\left(x \right)} = \frac{1}{5 x^{2} + 9}$$$:

$${\color{red}{\int{\frac{4}{3 \left(5 x^{2} + 9\right)} d x}}} = {\color{red}{\left(\frac{4 \int{\frac{1}{5 x^{2} + 9} d x}}{3}\right)}}$$

Zij $$$u=\frac{\sqrt{5}}{3} x$$$.

Dan $$$du=\left(\frac{\sqrt{5}}{3} x\right)^{\prime }dx = \frac{\sqrt{5}}{3} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{3 \sqrt{5} du}{5}$$$.

Dus,

$$\frac{4 {\color{red}{\int{\frac{1}{5 x^{2} + 9} d x}}}}{3} = \frac{4 {\color{red}{\int{\frac{\sqrt{5}}{15 \left(u^{2} + 1\right)} d u}}}}{3}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{\sqrt{5}}{15}$$$ en $$$f{\left(u \right)} = \frac{1}{u^{2} + 1}$$$:

$$\frac{4 {\color{red}{\int{\frac{\sqrt{5}}{15 \left(u^{2} + 1\right)} d u}}}}{3} = \frac{4 {\color{red}{\left(\frac{\sqrt{5} \int{\frac{1}{u^{2} + 1} d u}}{15}\right)}}}{3}$$

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

$$\frac{4 \sqrt{5} {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{45} = \frac{4 \sqrt{5} {\color{red}{\operatorname{atan}{\left(u \right)}}}}{45}$$

We herinneren eraan dat $$$u=\frac{\sqrt{5}}{3} x$$$:

$$\frac{4 \sqrt{5} \operatorname{atan}{\left({\color{red}{u}} \right)}}{45} = \frac{4 \sqrt{5} \operatorname{atan}{\left({\color{red}{\frac{\sqrt{5}}{3} x}} \right)}}{45}$$

Dus,

$$\int{\frac{4}{15 x^{2} + 27} d x} = \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} x}{3} \right)}}{45}$$

Voeg de integratieconstante toe:

$$\int{\frac{4}{15 x^{2} + 27} d x} = \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} x}{3} \right)}}{45}+C$$

$$$\int \frac{4}{15 x^{2} + 27}\, dx = \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} x}{3} \right)}}{45} + C$$$A