$$$\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1$$$の積分

$$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x}\right)}}$$

$$$c=1$$$ に対して定数則 $$$\int c\, dx = c x$$$ を適用する:

$$- \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{1 d x}}} = - \int{4 x^{3} d x} + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{x}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=4$$$ と $$$f{\left(x \right)} = x^{3}$$$ に対して適用する:

$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\int{4 x^{3} d x}}} = - x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - {\color{red}{\left(4 \int{x^{3} d x}\right)}}$$

$$$n=3$$$ 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:

$$- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\int{x^{3} d x}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- x + \int{\sqrt{2} x^{\frac{5}{2}} d x} - 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\sqrt{2}$$$ と $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$ に対して適用する:

$$- x^{4} - x + {\color{red}{\int{\sqrt{2} x^{\frac{5}{2}} d x}}} = - x^{4} - x + {\color{red}{\sqrt{2} \int{x^{\frac{5}{2}} d x}}}$$

$$$n=\frac{5}{2}$$$ 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:

$$- x^{4} - x + \sqrt{2} {\color{red}{\int{x^{\frac{5}{2}} d x}}}=- x^{4} - x + \sqrt{2} {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=- x^{4} - x + \sqrt{2} {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

したがって、

$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x$$

積分定数を加える:

$$\int{\left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x+C$$

$$$\int \left(\sqrt{2} x^{\frac{5}{2}} - 4 x^{3} - 1\right)\, dx = \left(\frac{2 \sqrt{2} x^{\frac{7}{2}}}{7} - x^{4} - x\right) + C$$$A