Integralen av $$$x^{4} - x^{2}$$$

Bestäm $$$\int \left(x^{4} - x^{2}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{4} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{x^{4} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=4$$$:

$$- \int{x^{2} d x} + {\color{red}{\int{x^{4} d x}}}=- \int{x^{2} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \int{x^{2} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

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

$$\frac{x^{5}}{5} - {\color{red}{\int{x^{2} d x}}}=\frac{x^{5}}{5} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{5}}{5} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Alltså,

$$\int{\left(x^{4} - x^{2}\right)d x} = \frac{x^{5}}{5} - \frac{x^{3}}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{4} - x^{2}\right)d x} = \frac{x^{5}}{5} - \frac{x^{3}}{3}+C$$

$$$\int \left(x^{4} - x^{2}\right)\, dx = \left(\frac{x^{5}}{5} - \frac{x^{3}}{3}\right) + C$$$A