$$$65536 x^{41}$$$の積分

$$$\int 65536 x^{41}\, dx$$$ 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、$$$c=65536$$$ と $$$f{\left(x \right)} = x^{41}$$$ に対して適用する:

$${\color{red}{\int{65536 x^{41} d x}}} = {\color{red}{\left(65536 \int{x^{41} d x}\right)}}$$

$$$n=41$$$ 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:

$$65536 {\color{red}{\int{x^{41} d x}}}=65536 {\color{red}{\frac{x^{1 + 41}}{1 + 41}}}=65536 {\color{red}{\left(\frac{x^{42}}{42}\right)}}$$

したがって、

$$\int{65536 x^{41} d x} = \frac{32768 x^{42}}{21}$$

積分定数を加える:

$$\int{65536 x^{41} d x} = \frac{32768 x^{42}}{21}+C$$

$$$\int 65536 x^{41}\, dx = \frac{32768 x^{42}}{21} + C$$$A