$$$\sinh^{2}{\left(x \right)}$$$ 的積分

求$$$\int \sinh^{2}{\left(x \right)}\, dx$$$。

套用降冪公式 $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$，令 $$$\alpha=x$$$:

$${\color{red}{\int{\sinh^{2}{\left(x \right)} d x}}} = {\color{red}{\int{\left(\frac{\cosh{\left(2 x \right)}}{2} - \frac{1}{2}\right)d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(x \right)} = \cosh{\left(2 x \right)} - 1$$$：

$${\color{red}{\int{\left(\frac{\cosh{\left(2 x \right)}}{2} - \frac{1}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(\cosh{\left(2 x \right)} - 1\right)d x}}{2}\right)}}$$

逐項積分：

$$\frac{{\color{red}{\int{\left(\cosh{\left(2 x \right)} - 1\right)d x}}}}{2} = \frac{{\color{red}{\left(- \int{1 d x} + \int{\cosh{\left(2 x \right)} d x}\right)}}}{2}$$

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

$$\frac{\int{\cosh{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{\int{1 d x}}}}{2} = \frac{\int{\cosh{\left(2 x \right)} d x}}{2} - \frac{{\color{red}{x}}}{2}$$

令 $$$u=2 x$$$。

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

該積分變為

$$- \frac{x}{2} + \frac{{\color{red}{\int{\cosh{\left(2 x \right)} d x}}}}{2} = - \frac{x}{2} + \frac{{\color{red}{\int{\frac{\cosh{\left(u \right)}}{2} d u}}}}{2}$$

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

$$- \frac{x}{2} + \frac{{\color{red}{\int{\frac{\cosh{\left(u \right)}}{2} d u}}}}{2} = - \frac{x}{2} + \frac{{\color{red}{\left(\frac{\int{\cosh{\left(u \right)} d u}}{2}\right)}}}{2}$$

雙曲餘弦的積分為 $$$\int{\cosh{\left(u \right)} d u} = \sinh{\left(u \right)}$$$：

$$- \frac{x}{2} + \frac{{\color{red}{\int{\cosh{\left(u \right)} d u}}}}{4} = - \frac{x}{2} + \frac{{\color{red}{\sinh{\left(u \right)}}}}{4}$$

回顧一下 $$$u=2 x$$$：

$$- \frac{x}{2} + \frac{\sinh{\left({\color{red}{u}} \right)}}{4} = - \frac{x}{2} + \frac{\sinh{\left({\color{red}{\left(2 x\right)}} \right)}}{4}$$

因此，

$$\int{\sinh^{2}{\left(x \right)} d x} = - \frac{x}{2} + \frac{\sinh{\left(2 x \right)}}{4}$$

加上積分常數：

$$\int{\sinh^{2}{\left(x \right)} d x} = - \frac{x}{2} + \frac{\sinh{\left(2 x \right)}}{4}+C$$

$$$\int \sinh^{2}{\left(x \right)}\, dx = \left(- \frac{x}{2} + \frac{\sinh{\left(2 x \right)}}{4}\right) + C$$$A