Integraal van $$$\frac{1458}{\sqrt{x^{3}}}$$$

Bepaal $$$\int \frac{1458}{\sqrt{x^{3}}}\, dx$$$.

De invoer is herschreven: $$$\int{\frac{1458}{\sqrt{x^{3}}} d x}=\int{\frac{1458}{x^{\frac{3}{2}}} d x}$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=1458$$$ en $$$f{\left(x \right)} = \frac{1}{x^{\frac{3}{2}}}$$$:

$${\color{red}{\int{\frac{1458}{x^{\frac{3}{2}}} d x}}} = {\color{red}{\left(1458 \int{\frac{1}{x^{\frac{3}{2}}} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{3}{2}$$$:

$$1458 {\color{red}{\int{\frac{1}{x^{\frac{3}{2}}} d x}}}=1458 {\color{red}{\int{x^{- \frac{3}{2}} d x}}}=1458 {\color{red}{\frac{x^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1}}}=1458 {\color{red}{\left(- 2 x^{- \frac{1}{2}}\right)}}=1458 {\color{red}{\left(- \frac{2}{\sqrt{x}}\right)}}$$

Dus,

$$\int{\frac{1458}{x^{\frac{3}{2}}} d x} = - \frac{2916}{\sqrt{x}}$$

Voeg de integratieconstante toe:

$$\int{\frac{1458}{x^{\frac{3}{2}}} d x} = - \frac{2916}{\sqrt{x}}+C$$

$$$\int \frac{1458}{\sqrt{x^{3}}}\, dx = - \frac{2916}{\sqrt{x}} + C$$$A