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

$$$\int \frac{\pi x^{2} \ln\left(3\right)}{e^{\pi}}\, dx$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{\pi \ln{\left(3 \right)}}{e^{\pi}}$$$ と $$$f{\left(x \right)} = x^{2}$$$ に対して適用する:

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

$$$n=2$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

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

したがって、

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

積分定数を加える:

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

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