$$$\frac{1}{x^{2} - 78 x}$$$ 的積分
您的輸入
求$$$\int \frac{1}{x^{2} - 78 x}\, dx$$$。
解答
進行部分分式分解（步驟可見 »）:
$${\color{red}{\int{\frac{1}{x^{2} - 78 x} d x}}} = {\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}}$$
逐項積分：
$${\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{78 x} d x} + \int{\frac{1}{78 \left(x - 78\right)} d x}\right)}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{78}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x}$$$：
$$\int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\int{\frac{1}{78 x} d x}}} = \int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\left(\frac{\int{\frac{1}{x} d x}}{78}\right)}}$$
$$$\frac{1}{x}$$$ 的積分是 $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$：
$$\int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\int{\frac{1}{x} d x}}}}{78} = \int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{78}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{78}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - 78}$$$：
$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\int{\frac{1}{78 \left(x - 78\right)} d x}}} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\left(\frac{\int{\frac{1}{x - 78} d x}}{78}\right)}}$$
令 $$$u=x - 78$$$。
則 $$$du=\left(x - 78\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。
該積分可改寫為
$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{x - 78} d x}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78}$$
$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：
$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{78}$$
回顧一下 $$$u=x - 78$$$：
$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{\left(x - 78\right)}}}\right| \right)}}{78}$$
因此，
$$\int{\frac{1}{x^{2} - 78 x} d x} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{x - 78}\right| \right)}}{78}$$
化簡：
$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}$$
加上積分常數：
$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}+C$$
答案
$$$\int \frac{1}{x^{2} - 78 x}\, dx = \frac{- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 78}\right|\right)}{78} + C$$$A