Integraal van $$$\frac{\pi x^{2} \ln\left(3\right)}{e^{\pi}}$$$

Bepaal $$$\int \frac{\pi x^{2} \ln\left(3\right)}{e^{\pi}}\, dx$$$.

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

$${\color{red}{\int{\frac{\pi x^{2} \ln{\left(3 \right)}}{e^{\pi}} d x}}} = {\color{red}{\frac{\pi \ln{\left(3 \right)} \int{x^{2} d x}}{e^{\pi}}}}$$

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

$$\frac{\pi \ln{\left(3 \right)} {\color{red}{\int{x^{2} d x}}}}{e^{\pi}}=\frac{\pi \ln{\left(3 \right)} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{e^{\pi}}=\frac{\pi \ln{\left(3 \right)} {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{e^{\pi}}$$

Dus,

$$\int{\frac{\pi x^{2} \ln{\left(3 \right)}}{e^{\pi}} d x} = \frac{\pi x^{3} \ln{\left(3 \right)}}{3 e^{\pi}}$$

Voeg de integratieconstante toe:

$$\int{\frac{\pi x^{2} \ln{\left(3 \right)}}{e^{\pi}} d x} = \frac{\pi x^{3} \ln{\left(3 \right)}}{3 e^{\pi}}+C$$

$$$\int \frac{\pi x^{2} \ln\left(3\right)}{e^{\pi}}\, dx = \frac{\pi x^{3} \ln\left(3\right)}{3 e^{\pi}} + C$$$A