$$$\frac{1}{x^{4} - 5}$$$ 的積分

求$$$\int \frac{1}{x^{4} - 5}\, dx$$$。

進行部分分式分解（步驟可見 »）:

$${\color{red}{\int{\frac{1}{x^{4} - 5} d x}}} = {\color{red}{\int{\left(- \frac{\sqrt{5}}{10 \left(x^{2} + \sqrt{5}\right)} - \frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} + \frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)}\right)d x}}}$$

逐項積分：

$${\color{red}{\int{\left(- \frac{\sqrt{5}}{10 \left(x^{2} + \sqrt{5}\right)} - \frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} + \frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)}\right)d x}}} = {\color{red}{\left(\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt{5}}{10 \left(x^{2} + \sqrt{5}\right)} d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{\sqrt{5}}{10}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x^{2} + \sqrt{5}}$$$：

$$\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - {\color{red}{\int{\frac{\sqrt{5}}{10 \left(x^{2} + \sqrt{5}\right)} d x}}} = \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - {\color{red}{\left(\frac{\sqrt{5} \int{\frac{1}{x^{2} + \sqrt{5}} d x}}{10}\right)}}$$

令 $$$u=\frac{5^{\frac{3}{4}}}{5} x$$$。

則 $$$du=\left(\frac{5^{\frac{3}{4}}}{5} x\right)^{\prime }dx = \frac{5^{\frac{3}{4}}}{5} dx$$$ (步驟見»)，並可得 $$$dx = \sqrt[4]{5} du$$$。

因此，

$$\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt{5} {\color{red}{\int{\frac{1}{x^{2} + \sqrt{5}} d x}}}}{10} = \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt{5} {\color{red}{\int{\frac{5^{\frac{3}{4}}}{5 \left(u^{2} + 1\right)} d u}}}}{10}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{5^{\frac{3}{4}}}{5}$$$ 與 $$$f{\left(u \right)} = \frac{1}{u^{2} + 1}$$$：

$$\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt{5} {\color{red}{\int{\frac{5^{\frac{3}{4}}}{5 \left(u^{2} + 1\right)} d u}}}}{10} = \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt{5} {\color{red}{\left(\frac{5^{\frac{3}{4}} \int{\frac{1}{u^{2} + 1} d u}}{5}\right)}}}{10}$$

$$$\frac{1}{u^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$：

$$\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{10} = \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\operatorname{atan}{\left(u \right)}}}}{10}$$

回顧一下 $$$u=\frac{5^{\frac{3}{4}}}{5} x$$$：

$$\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} \operatorname{atan}{\left({\color{red}{u}} \right)}}{10} = \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} \operatorname{atan}{\left({\color{red}{\frac{5^{\frac{3}{4}}}{5} x}} \right)}}{10}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{\sqrt[4]{5}}{20}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x + \sqrt[4]{5}}$$$：

$$- \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - {\color{red}{\int{\frac{\sqrt[4]{5}}{20 \left(x + \sqrt[4]{5}\right)} d x}}} = - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - {\color{red}{\left(\frac{\sqrt[4]{5} \int{\frac{1}{x + \sqrt[4]{5}} d x}}{20}\right)}}$$

令 $$$u=x + \sqrt[4]{5}$$$。

則 $$$du=\left(x + \sqrt[4]{5}\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

該積分變為

$$- \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{x + \sqrt[4]{5}} d x}}}}{20} = - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{u} d u}}}}{20}$$

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

$$- \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{u} d u}}}}{20} = - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} - \frac{\sqrt[4]{5} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{20}$$

回顧一下 $$$u=x + \sqrt[4]{5}$$$：

$$- \frac{\sqrt[4]{5} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x} = - \frac{\sqrt[4]{5} \ln{\left(\left|{{\color{red}{\left(x + \sqrt[4]{5}\right)}}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{\sqrt[4]{5}}{20}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - \sqrt[4]{5}}$$$：

$$- \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + {\color{red}{\int{\frac{\sqrt[4]{5}}{20 \left(x - \sqrt[4]{5}\right)} d x}}} = - \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + {\color{red}{\left(\frac{\sqrt[4]{5} \int{\frac{1}{x - \sqrt[4]{5}} d x}}{20}\right)}}$$

令 $$$u=x - \sqrt[4]{5}$$$。

則 $$$du=\left(x - \sqrt[4]{5}\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

該積分可改寫為

$$- \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{x - \sqrt[4]{5}} d x}}}}{20} = - \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{u} d u}}}}{20}$$

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

$$- \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \frac{\sqrt[4]{5} {\color{red}{\int{\frac{1}{u} d u}}}}{20} = - \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} + \frac{\sqrt[4]{5} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{20}$$

回顧一下 $$$u=x - \sqrt[4]{5}$$$：

$$- \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} + \frac{\sqrt[4]{5} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10} = - \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} + \frac{\sqrt[4]{5} \ln{\left(\left|{{\color{red}{\left(x - \sqrt[4]{5}\right)}}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}}}{5} x \right)}}{10}$$

因此，

$$\int{\frac{1}{x^{4} - 5} d x} = \frac{\sqrt[4]{5} \ln{\left(\left|{x - \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)}}{20} - \frac{\sqrt[4]{5} \operatorname{atan}{\left(\frac{5^{\frac{3}{4}} x}{5} \right)}}{10}$$

化簡：

$$\int{\frac{1}{x^{4} - 5} d x} = \frac{\sqrt[4]{5} \left(\ln{\left(\left|{x - \sqrt[4]{5}}\right| \right)} - \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)} - 2 \operatorname{atan}{\left(\frac{5^{\frac{3}{4}} x}{5} \right)}\right)}{20}$$

加上積分常數：

$$\int{\frac{1}{x^{4} - 5} d x} = \frac{\sqrt[4]{5} \left(\ln{\left(\left|{x - \sqrt[4]{5}}\right| \right)} - \ln{\left(\left|{x + \sqrt[4]{5}}\right| \right)} - 2 \operatorname{atan}{\left(\frac{5^{\frac{3}{4}} x}{5} \right)}\right)}{20}+C$$

$$$\int \frac{1}{x^{4} - 5}\, dx = \frac{\sqrt[4]{5} \left(\ln\left(\left|{x - \sqrt[4]{5}}\right|\right) - \ln\left(\left|{x + \sqrt[4]{5}}\right|\right) - 2 \operatorname{atan}{\left(\frac{5^{\frac{3}{4}} x}{5} \right)}\right)}{20} + C$$$A