Integralen av $$$\frac{x^{2}}{7 - x^{3}}$$$

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

Låt $$$u=7 - x^{3}$$$ vara.

Då $$$du=\left(7 - x^{3}\right)^{\prime }dx = - 3 x^{2} dx$$$ (stegen kan ses »), och vi har att $$$x^{2} dx = - \frac{du}{3}$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=- \frac{1}{3}$$$ och $$$f{\left(u \right)} = \frac{1}{u}$$$:

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

Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Kom ihåg att $$$u=7 - x^{3}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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