Integraal van $$$\frac{1}{- c + c_{max}}$$$ met betrekking tot $$$c$$$

Bepaal $$$\int \frac{1}{- c + c_{max}}\, dc$$$.

Zij $$$u=- c + c_{max}$$$.

Dan $$$du=\left(- c + c_{max}\right)^{\prime }dc = - dc$$$ (de stappen zijn te zien »), en dan geldt dat $$$dc = - du$$$.

Dus,

$${\color{red}{\int{\frac{1}{- c + c_{max}} d c}}} = {\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=-1$$$ en $$$f{\left(u \right)} = \frac{1}{u}$$$:

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

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 $$$u=- c + c_{max}$$$:

$$- \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\left(- c + c_{max}\right)}}}\right| \right)}$$

Dus,

$$\int{\frac{1}{- c + c_{max}} d c} = - \ln{\left(\left|{c - c_{max}}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{- c + c_{max}} d c} = - \ln{\left(\left|{c - c_{max}}\right| \right)}+C$$

$$$\int \frac{1}{- c + c_{max}}\, dc = - \ln\left(\left|{c - c_{max}}\right|\right) + C$$$A