$$$\frac{1}{a - p}$$$ 关于$$$a$$$的积分

求$$$\int \frac{1}{a - p}\, da$$$。

设$$$u=a - p$$$。

则$$$du=\left(a - p\right)^{\prime }da = 1 da$$$ (步骤见»)，并有$$$da = du$$$。

所以，

$${\color{red}{\int{\frac{1}{a - p} d a}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回忆一下 $$$u=a - p$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\left(a - p\right)}}}\right| \right)}$$

因此，

$$\int{\frac{1}{a - p} d a} = \ln{\left(\left|{a - p}\right| \right)}$$

加上积分常数：

$$\int{\frac{1}{a - p} d a} = \ln{\left(\left|{a - p}\right| \right)}+C$$

$$$\int \frac{1}{a - p}\, da = \ln\left(\left|{a - p}\right|\right) + C$$$A