Integralen av $$$- \frac{7 x^{2}}{8}$$$

Bestäm $$$\int \left(- \frac{7 x^{2}}{8}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{7}{8}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\left(- \frac{7 x^{2}}{8}\right)d x}}} = {\color{red}{\left(- \frac{7 \int{x^{2} d x}}{8}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

$$- \frac{7 {\color{red}{\int{x^{2} d x}}}}{8}=- \frac{7 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{8}=- \frac{7 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{8}$$

Alltså,

$$\int{\left(- \frac{7 x^{2}}{8}\right)d x} = - \frac{7 x^{3}}{24}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{7 x^{2}}{8}\right)d x} = - \frac{7 x^{3}}{24}+C$$

$$$\int \left(- \frac{7 x^{2}}{8}\right)\, dx = - \frac{7 x^{3}}{24} + C$$$A