Integraal van $$$\frac{x^{6} \sqrt{x^{5}}}{\sqrt{x^{7}}}$$$

Bepaal $$$\int \frac{x^{6} \sqrt{x^{5}}}{\sqrt{x^{7}}}\, dx$$$.

De invoer is herschreven: $$$\int{\frac{x^{6} \sqrt{x^{5}}}{\sqrt{x^{7}}} d x}=\int{x^{5} d x}$$$.

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

$${\color{red}{\int{x^{5} d x}}}={\color{red}{\frac{x^{1 + 5}}{1 + 5}}}={\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Dus,

$$\int{x^{5} d x} = \frac{x^{6}}{6}$$

Voeg de integratieconstante toe:

$$\int{x^{5} d x} = \frac{x^{6}}{6}+C$$

$$$\int \frac{x^{6} \sqrt{x^{5}}}{\sqrt{x^{7}}}\, dx = \frac{x^{6}}{6} + C$$$A