$$$\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}$$$ 的积分

求$$$\int \frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}\, dx$$$。

输入已重写为：$$$\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}=\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x}$$$。

化简被积函数:

$${\color{red}{\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x}}} = {\color{red}{\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}}}$$

对 $$$c=\frac{\sqrt{7}}{7}$$$ 和 $$$f{\left(x \right)} = \frac{\sqrt{7 - x}}{\sqrt{x}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}}} = {\color{red}{\left(\frac{\sqrt{7} \int{\frac{\sqrt{7 - x}}{\sqrt{x}} d x}}{7}\right)}}$$

设$$$u=\sqrt{x}$$$。

则$$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (步骤见»)，并有$$$\frac{dx}{\sqrt{x}} = 2 du$$$。

积分变为

$$\frac{\sqrt{7} {\color{red}{\int{\frac{\sqrt{7 - x}}{\sqrt{x}} d x}}}}{7} = \frac{\sqrt{7} {\color{red}{\int{2 \sqrt{7 - u^{2}} d u}}}}{7}$$

对 $$$c=2$$$ 和 $$$f{\left(u \right)} = \sqrt{7 - u^{2}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{\sqrt{7} {\color{red}{\int{2 \sqrt{7 - u^{2}} d u}}}}{7} = \frac{\sqrt{7} {\color{red}{\left(2 \int{\sqrt{7 - u^{2}} d u}\right)}}}{7}$$

设$$$u=\sqrt{7} \sin{\left(v \right)}$$$。

则$$$du=\left(\sqrt{7} \sin{\left(v \right)}\right)^{\prime }dv = \sqrt{7} \cos{\left(v \right)} dv$$$（步骤见»）。

此外，可得$$$v=\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}$$$。

因此，

$$$\sqrt{7 - u ^{2}} = \sqrt{7 - 7 \sin^{2}{\left( v \right)}}$$$

利用恒等式 $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$：

$$$\sqrt{7 - 7 \sin^{2}{\left( v \right)}}=\sqrt{7} \sqrt{1 - \sin^{2}{\left( v \right)}}=\sqrt{7} \sqrt{\cos^{2}{\left( v \right)}}$$$

假设$$$\cos{\left( v \right)} \ge 0$$$，我们得到如下结果：

$$$\sqrt{7} \sqrt{\cos^{2}{\left( v \right)}} = \sqrt{7} \cos{\left( v \right)}$$$

因此，

$$\frac{2 \sqrt{7} {\color{red}{\int{\sqrt{7 - u^{2}} d u}}}}{7} = \frac{2 \sqrt{7} {\color{red}{\int{7 \cos^{2}{\left(v \right)} d v}}}}{7}$$

对 $$$c=7$$$ 和 $$$f{\left(v \right)} = \cos^{2}{\left(v \right)}$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：

$$\frac{2 \sqrt{7} {\color{red}{\int{7 \cos^{2}{\left(v \right)} d v}}}}{7} = \frac{2 \sqrt{7} {\color{red}{\left(7 \int{\cos^{2}{\left(v \right)} d v}\right)}}}{7}$$

应用降幂公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，并令 $$$\alpha= v $$$:

$$2 \sqrt{7} {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}} = 2 \sqrt{7} {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：

$$2 \sqrt{7} {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}} = 2 \sqrt{7} {\color{red}{\left(\frac{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}{2}\right)}}$$

逐项积分：

$$\sqrt{7} {\color{red}{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}} = \sqrt{7} {\color{red}{\left(\int{1 d v} + \int{\cos{\left(2 v \right)} d v}\right)}}$$

应用常数法则 $$$\int c\, dv = c v$$$，使用 $$$c=1$$$：

$$\sqrt{7} \left(\int{\cos{\left(2 v \right)} d v} + {\color{red}{\int{1 d v}}}\right) = \sqrt{7} \left(\int{\cos{\left(2 v \right)} d v} + {\color{red}{v}}\right)$$

设$$$w=2 v$$$。

则$$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (步骤见»)，并有$$$dv = \frac{dw}{2}$$$。

积分变为

$$\sqrt{7} \left(v + {\color{red}{\int{\cos{\left(2 v \right)} d v}}}\right) = \sqrt{7} \left(v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}\right)$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(w \right)} = \cos{\left(w \right)}$$$ 应用常数倍法则 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$：

$$\sqrt{7} \left(v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}\right) = \sqrt{7} \left(v + {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}\right)$$

余弦函数的积分为 $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$：

$$\sqrt{7} \left(v + \frac{{\color{red}{\int{\cos{\left(w \right)} d w}}}}{2}\right) = \sqrt{7} \left(v + \frac{{\color{red}{\sin{\left(w \right)}}}}{2}\right)$$

回忆一下 $$$w=2 v$$$:

$$\sqrt{7} \left(v + \frac{\sin{\left({\color{red}{w}} \right)}}{2}\right) = \sqrt{7} \left(v + \frac{\sin{\left({\color{red}{\left(2 v\right)}} \right)}}{2}\right)$$

回忆一下 $$$v=\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}$$$:

$$\sqrt{7} \left(\frac{\sin{\left(2 {\color{red}{v}} \right)}}{2} + {\color{red}{v}}\right) = \sqrt{7} \left(\frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}}} \right)}}{2} + {\color{red}{\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}}}\right)$$

回忆一下 $$$u=\sqrt{x}$$$:

$$\sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{u}}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{u}}}{7} \right)}\right) = \sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{\sqrt{x}}}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} {\color{red}{\sqrt{x}}}}{7} \right)}\right)$$

因此，

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)} \right)}}{2} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

使用公式 $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$，化简该表达式：

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{7} \sqrt{x} \sqrt{1 - \frac{x}{7}}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

进一步化简：

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)$$

加上积分常数：

$$\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x} = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right)+C$$

$$$\int \frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}\, dx = \sqrt{7} \left(\frac{\sqrt{x} \sqrt{7 - x}}{7} + \operatorname{asin}{\left(\frac{\sqrt{7} \sqrt{x}}{7} \right)}\right) + C$$$A