Integraal van $$$\frac{x^{2}}{7 - x^{3}}$$$

Bepaal $$$\int \frac{x^{2}}{7 - x^{3}}\, dx$$$.

Zij $$$u=7 - x^{3}$$$.

Dan $$$du=\left(7 - x^{3}\right)^{\prime }dx = - 3 x^{2} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$x^{2} dx = - \frac{du}{3}$$$.

Dus,

$${\color{red}{\int{\frac{x^{2}}{7 - x^{3}} d x}}} = {\color{red}{\int{\left(- \frac{1}{3 u}\right)d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=- \frac{1}{3}$$$ en $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\left(- \frac{1}{3 u}\right)d u}}} = {\color{red}{\left(- \frac{\int{\frac{1}{u} d u}}{3}\right)}}$$

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

$$- \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{3} = - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{3}$$

We herinneren eraan dat $$$u=7 - x^{3}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{3} = - \frac{\ln{\left(\left|{{\color{red}{\left(7 - x^{3}\right)}}}\right| \right)}}{3}$$

Dus,

$$\int{\frac{x^{2}}{7 - x^{3}} d x} = - \frac{\ln{\left(\left|{x^{3} - 7}\right| \right)}}{3}$$

Voeg de integratieconstante toe:

$$\int{\frac{x^{2}}{7 - x^{3}} d x} = - \frac{\ln{\left(\left|{x^{3} - 7}\right| \right)}}{3}+C$$

$$$\int \frac{x^{2}}{7 - x^{3}}\, dx = - \frac{\ln\left(\left|{x^{3} - 7}\right|\right)}{3} + C$$$A