$$$i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1$$$ 對 $$$x$$$ 的積分

求$$$\int \left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)\, dx$$$。

已將輸入重寫為：$$$\int{\left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)d x}=\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x}$$$。

逐項積分：

$${\color{red}{\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$- \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{\int{1 d x}}} = - \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{x}}$$

配合 $$$c=y^{2}$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$- x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{\int{y^{2} d x}}} = - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{x y^{2}}}$$

配合 $$$c=i n t$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$- x y^{2} - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + {\color{red}{\int{i n t d x}}} = - x y^{2} - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + {\color{red}{i n t x}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\sqrt{2} y^{2}$$$ 與 $$$f{\left(x \right)} = x^{\frac{7}{2}}$$$：

$$i n t x - x y^{2} - x - {\color{red}{\int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x}}} = i n t x - x y^{2} - x - {\color{red}{\sqrt{2} y^{2} \int{x^{\frac{7}{2}} d x}}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=\frac{7}{2}$$$：

$$i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\int{x^{\frac{7}{2}} d x}}}=i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\frac{x^{1 + \frac{7}{2}}}{1 + \frac{7}{2}}}}=i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\left(\frac{2 x^{\frac{9}{2}}}{9}\right)}}$$

因此，

$$\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x} = i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x$$

加上積分常數：

$$\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x} = i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x+C$$

$$$\int \left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)\, dx = \left(i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x\right) + C$$$A