Integralen av $$$3 x^{3}$$$

Bestäm $$$\int 3 x^{3}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = x^{3}$$$:

$${\color{red}{\int{3 x^{3} d x}}} = {\color{red}{\left(3 \int{x^{3} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=3$$$:

$$3 {\color{red}{\int{x^{3} d x}}}=3 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=3 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Alltså,

$$\int{3 x^{3} d x} = \frac{3 x^{4}}{4}$$

Lägg till integrationskonstanten:

$$\int{3 x^{3} d x} = \frac{3 x^{4}}{4}+C$$

$$$\int 3 x^{3}\, dx = \frac{3 x^{4}}{4} + C$$$A