$$$\sqrt{\frac{a - x}{x}}$$$ 关于$$$x$$$的积分

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

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

设$$$u=\sqrt{x}$$$。

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

所以，

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

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

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

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

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

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

因此，

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

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

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

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

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

因此，

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

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

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

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

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

Expand the expression:

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

逐项积分：

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

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

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

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

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

设$$$w=2 v$$$。

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

因此，

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

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

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

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

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

回忆一下 $$$w=2 v$$$:

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

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

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

回忆一下 $$$u=\sqrt{x}$$$:

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

因此，

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

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：

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

进一步化简：

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

加上积分常数：

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

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