Integralen av $$$243 x^{10}$$$

Bestäm $$$\int 243 x^{10}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=243$$$ och $$$f{\left(x \right)} = x^{10}$$$:

$${\color{red}{\int{243 x^{10} d x}}} = {\color{red}{\left(243 \int{x^{10} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=10$$$:

$$243 {\color{red}{\int{x^{10} d x}}}=243 {\color{red}{\frac{x^{1 + 10}}{1 + 10}}}=243 {\color{red}{\left(\frac{x^{11}}{11}\right)}}$$

Alltså,

$$\int{243 x^{10} d x} = \frac{243 x^{11}}{11}$$

Lägg till integrationskonstanten:

$$\int{243 x^{10} d x} = \frac{243 x^{11}}{11}+C$$

$$$\int 243 x^{10}\, dx = \frac{243 x^{11}}{11} + C$$$A