Integraal van $$$\frac{7}{\sqrt{x^{5}}}$$$

Bepaal $$$\int \frac{7}{\sqrt{x^{5}}}\, dx$$$.

De invoer is herschreven: $$$\int{\frac{7}{\sqrt{x^{5}}} d x}=\int{\frac{7}{x^{\frac{5}{2}}} d x}$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=7$$$ en $$$f{\left(x \right)} = \frac{1}{x^{\frac{5}{2}}}$$$:

$${\color{red}{\int{\frac{7}{x^{\frac{5}{2}}} d x}}} = {\color{red}{\left(7 \int{\frac{1}{x^{\frac{5}{2}}} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{5}{2}$$$:

$$7 {\color{red}{\int{\frac{1}{x^{\frac{5}{2}}} d x}}}=7 {\color{red}{\int{x^{- \frac{5}{2}} d x}}}=7 {\color{red}{\frac{x^{- \frac{5}{2} + 1}}{- \frac{5}{2} + 1}}}=7 {\color{red}{\left(- \frac{2 x^{- \frac{3}{2}}}{3}\right)}}=7 {\color{red}{\left(- \frac{2}{3 x^{\frac{3}{2}}}\right)}}$$

Dus,

$$\int{\frac{7}{x^{\frac{5}{2}}} d x} = - \frac{14}{3 x^{\frac{3}{2}}}$$

Voeg de integratieconstante toe:

$$\int{\frac{7}{x^{\frac{5}{2}}} d x} = - \frac{14}{3 x^{\frac{3}{2}}}+C$$

$$$\int \frac{7}{\sqrt{x^{5}}}\, dx = - \frac{14}{3 x^{\frac{3}{2}}} + C$$$A