Integraal van $$$\frac{x^{3}}{2 \sqrt{x^{5}}}$$$

Bepaal $$$\int \frac{x^{3}}{2 \sqrt{x^{5}}}\, dx$$$.

De invoer is herschreven: $$$\int{\frac{x^{3}}{2 \sqrt{x^{5}}} d x}=\int{\frac{\sqrt{x}}{2} d x}$$$.

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

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

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

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

Dus,

$$\int{\frac{\sqrt{x}}{2} d x} = \frac{x^{\frac{3}{2}}}{3}$$

Voeg de integratieconstante toe:

$$\int{\frac{\sqrt{x}}{2} d x} = \frac{x^{\frac{3}{2}}}{3}+C$$

$$$\int \frac{x^{3}}{2 \sqrt{x^{5}}}\, dx = \frac{x^{\frac{3}{2}}}{3} + C$$$A