$$$\sqrt{\frac{1 - t}{t}}$$$ 的積分

求$$$\int \sqrt{\frac{1 - t}{t}}\, dt$$$。

已將輸入重寫為：$$$\int{\sqrt{\frac{1 - t}{t}} d t}=\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t}$$$。

令 $$$u=\sqrt{t}$$$。

則 $$$du=\left(\sqrt{t}\right)^{\prime }dt = \frac{1}{2 \sqrt{t}} dt$$$ (步驟見»)，並可得 $$$\frac{dt}{\sqrt{t}} = 2 du$$$。

該積分變為

$${\color{red}{\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t}}} = {\color{red}{\int{2 \sqrt{1 - u^{2}} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=2$$$ 與 $$$f{\left(u \right)} = \sqrt{1 - u^{2}}$$$：

$${\color{red}{\int{2 \sqrt{1 - u^{2}} d u}}} = {\color{red}{\left(2 \int{\sqrt{1 - u^{2}} d u}\right)}}$$

令 $$$u=\sin{\left(v \right)}$$$。

則 $$$du=\left(\sin{\left(v \right)}\right)^{\prime }dv = \cos{\left(v \right)} dv$$$（步驟見»）。

此外，由此可得 $$$v=\operatorname{asin}{\left(u \right)}$$$。

所以，

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

使用恆等式 $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$：

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

假設 $$$\cos{\left( v \right)} \ge 0$$$，可得如下：

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

所以，

$$2 {\color{red}{\int{\sqrt{1 - u^{2}} d u}}} = 2 {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}}$$

套用降冪公式 $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，令 $$$\alpha= v $$$:

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

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$：

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

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dv = c v$$$：

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

令 $$$w=2 v$$$。

則 $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (步驟見»)，並可得 $$$dv = \frac{dw}{2}$$$。

因此，

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

套用常數倍法則 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(w \right)} = \cos{\left(w \right)}$$$：

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

餘弦函數的積分為 $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$：

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

回顧一下 $$$w=2 v$$$：

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

回顧一下 $$$v=\operatorname{asin}{\left(u \right)}$$$：

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

回顧一下 $$$u=\sqrt{t}$$$：

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

因此，

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \frac{\sin{\left(2 \operatorname{asin}{\left(\sqrt{t} \right)} \right)}}{2} + \operatorname{asin}{\left(\sqrt{t} \right)}$$

使用公式 $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$，化簡該表達式：

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

加上積分常數：

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

$$$\int \sqrt{\frac{1 - t}{t}}\, dt = \left(\sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}\right) + C$$$A