Integraal van $$$\frac{1}{- a + t}$$$ met betrekking tot $$$t$$$

Bepaal $$$\int \frac{1}{- a + t}\, dt$$$.

Zij $$$u=- a + t$$$.

Dan $$$du=\left(- a + t\right)^{\prime }dt = 1 dt$$$ (de stappen zijn te zien »), en dan geldt dat $$$dt = du$$$.

Dus,

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

De integraal van $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

We herinneren eraan dat $$$u=- a + t$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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