Integraal van $$$\frac{11}{\sqrt{t}}$$$

Bepaal $$$\int \frac{11}{\sqrt{t}}\, dt$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ toe met $$$c=11$$$ en $$$f{\left(t \right)} = \frac{1}{\sqrt{t}}$$$:

$${\color{red}{\int{\frac{11}{\sqrt{t}} d t}}} = {\color{red}{\left(11 \int{\frac{1}{\sqrt{t}} d t}\right)}}$$

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

$$11 {\color{red}{\int{\frac{1}{\sqrt{t}} d t}}}=11 {\color{red}{\int{t^{- \frac{1}{2}} d t}}}=11 {\color{red}{\frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}=11 {\color{red}{\left(2 t^{\frac{1}{2}}\right)}}=11 {\color{red}{\left(2 \sqrt{t}\right)}}$$

Dus,

$$\int{\frac{11}{\sqrt{t}} d t} = 22 \sqrt{t}$$

Voeg de integratieconstante toe:

$$\int{\frac{11}{\sqrt{t}} d t} = 22 \sqrt{t}+C$$

$$$\int \frac{11}{\sqrt{t}}\, dt = 22 \sqrt{t} + C$$$A