$$$\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}}$$$ 的積分

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

已將輸入重寫為：$$$\int{\frac{\sqrt{7} \sqrt{7 - x}}{7 \sqrt{x}} d x}=\int{\frac{\sqrt{49 - 7 x}}{7 \sqrt{x}} d x}$$$。

簡化被積函數:

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

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

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

令 $$$u=\sqrt{x}$$$。

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

該積分可改寫為

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

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 $$$f{\left(u \right)} = \sqrt{7 - u^{2}}$$$：

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

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

則 $$$du=\left(\sqrt{7} \sin{\left(v \right)}\right)^{\prime }dv = \sqrt{7} \cos{\left(v \right)} dv$$$（步驟見»）。

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

所以，

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

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

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

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

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

積分可以改寫為

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

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=7$$$ 與 $$$f{\left(v \right)} = \cos^{2}{\left(v \right)}$$$：

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

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

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

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 $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$：

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

逐項積分：

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

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

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

令 $$$w=2 v$$$。

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

所以，

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

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 $$$f{\left(w \right)} = \cos{\left(w \right)}$$$：

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

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

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

回顧一下 $$$w=2 v$$$：

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

回顧一下 $$$v=\operatorname{asin}{\left(\frac{\sqrt{7} u}{7} \right)}$$$：

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

回顧一下 $$$u=\sqrt{x}$$$：

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

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

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

進一步化簡：

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

加上積分常數：

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

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