Integraal van $$$- \frac{1}{\sqrt{u}}$$$

Bepaal $$$\int \left(- \frac{1}{\sqrt{u}}\right)\, du$$$.

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

$${\color{red}{\int{\left(- \frac{1}{\sqrt{u}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{u}} d u}\right)}}$$

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

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

Dus,

$$\int{\left(- \frac{1}{\sqrt{u}}\right)d u} = - 2 \sqrt{u}$$

Voeg de integratieconstante toe:

$$\int{\left(- \frac{1}{\sqrt{u}}\right)d u} = - 2 \sqrt{u}+C$$

$$$\int \left(- \frac{1}{\sqrt{u}}\right)\, du = - 2 \sqrt{u} + C$$$A