Integraal van $$$3 x^{\pi}$$$

Bepaal $$$\int 3 x^{\pi}\, dx$$$.

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

$${\color{red}{\int{3 x^{\pi} d x}}} = {\color{red}{\left(3 \int{x^{\pi} d x}\right)}}$$

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

$$3 {\color{red}{\int{x^{\pi} d x}}}=3 {\color{red}{\frac{x^{1 + \pi}}{1 + \pi}}}=3 {\color{red}{\frac{x^{1 + \pi}}{1 + \pi}}}$$

Dus,

$$\int{3 x^{\pi} d x} = \frac{3 x^{1 + \pi}}{1 + \pi}$$

Voeg de integratieconstante toe:

$$\int{3 x^{\pi} d x} = \frac{3 x^{1 + \pi}}{1 + \pi}+C$$

$$$\int 3 x^{\pi}\, dx = \frac{3 x^{1 + \pi}}{1 + \pi} + C$$$A