$$$5 x^{38} \left(6 x^{3} - 9\right)$$$の積分

$$$\int 5 x^{38} \left(6 x^{3} - 9\right)\, dx$$$ を求めよ。

入力は次のように書き換えられます: $$$\int{5 x^{38} \left(6 x^{3} - 9\right) d x}=\int{x^{38} \left(30 x^{3} - 45\right) d x}$$$。

被積分関数を簡単化する:

$${\color{red}{\int{x^{38} \left(30 x^{3} - 45\right) d x}}} = {\color{red}{\int{15 x^{38} \left(2 x^{3} - 3\right) d x}}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=15$$$ と $$$f{\left(x \right)} = x^{38} \left(2 x^{3} - 3\right)$$$ に対して適用する:

$${\color{red}{\int{15 x^{38} \left(2 x^{3} - 3\right) d x}}} = {\color{red}{\left(15 \int{x^{38} \left(2 x^{3} - 3\right) d x}\right)}}$$

Expand the expression:

$$15 {\color{red}{\int{x^{38} \left(2 x^{3} - 3\right) d x}}} = 15 {\color{red}{\int{\left(2 x^{41} - 3 x^{38}\right)d x}}}$$

項別に積分せよ:

$$15 {\color{red}{\int{\left(2 x^{41} - 3 x^{38}\right)d x}}} = 15 {\color{red}{\left(- \int{3 x^{38} d x} + \int{2 x^{41} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=3$$$ と $$$f{\left(x \right)} = x^{38}$$$ に対して適用する:

$$15 \int{2 x^{41} d x} - 15 {\color{red}{\int{3 x^{38} d x}}} = 15 \int{2 x^{41} d x} - 15 {\color{red}{\left(3 \int{x^{38} d x}\right)}}$$

$$$n=38$$$ 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:

$$15 \int{2 x^{41} d x} - 45 {\color{red}{\int{x^{38} d x}}}=15 \int{2 x^{41} d x} - 45 {\color{red}{\frac{x^{1 + 38}}{1 + 38}}}=15 \int{2 x^{41} d x} - 45 {\color{red}{\left(\frac{x^{39}}{39}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=2$$$ と $$$f{\left(x \right)} = x^{41}$$$ に対して適用する:

$$- \frac{15 x^{39}}{13} + 15 {\color{red}{\int{2 x^{41} d x}}} = - \frac{15 x^{39}}{13} + 15 {\color{red}{\left(2 \int{x^{41} d x}\right)}}$$

$$$n=41$$$ 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:

$$- \frac{15 x^{39}}{13} + 30 {\color{red}{\int{x^{41} d x}}}=- \frac{15 x^{39}}{13} + 30 {\color{red}{\frac{x^{1 + 41}}{1 + 41}}}=- \frac{15 x^{39}}{13} + 30 {\color{red}{\left(\frac{x^{42}}{42}\right)}}$$

したがって、

$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{42}}{7} - \frac{15 x^{39}}{13}$$

簡単化せよ:

$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91}$$

積分定数を加える:

$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91}+C$$

$$$\int 5 x^{38} \left(6 x^{3} - 9\right)\, dx = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91} + C$$$A