Integral von $$$\frac{1}{a - p}$$$ nach $$$a$$$

Bestimme $$$\int \frac{1}{a - p}\, da$$$.

Sei $$$u=a - p$$$.

Dann $$$du=\left(a - p\right)^{\prime }da = 1 da$$$ (die Schritte sind » zu sehen), und es gilt $$$da = du$$$.

Also,

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

Das Integral von $$$\frac{1}{u}$$$ ist $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Zur Erinnerung: $$$u=a - p$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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