Integraal van $$$- \sqrt{x}$$$

Bepaal $$$\int \left(- \sqrt{x}\right)\, dx$$$.

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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