Integralen av $$$\frac{5}{x^{66}}$$$

Bestäm $$$\int \frac{5}{x^{66}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = \frac{1}{x^{66}}$$$:

$${\color{red}{\int{\frac{5}{x^{66}} d x}}} = {\color{red}{\left(5 \int{\frac{1}{x^{66}} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-66$$$:

$$5 {\color{red}{\int{\frac{1}{x^{66}} d x}}}=5 {\color{red}{\int{x^{-66} d x}}}=5 {\color{red}{\frac{x^{-66 + 1}}{-66 + 1}}}=5 {\color{red}{\left(- \frac{x^{-65}}{65}\right)}}=5 {\color{red}{\left(- \frac{1}{65 x^{65}}\right)}}$$

Alltså,

$$\int{\frac{5}{x^{66}} d x} = - \frac{1}{13 x^{65}}$$

Lägg till integrationskonstanten:

$$\int{\frac{5}{x^{66}} d x} = - \frac{1}{13 x^{65}}+C$$

$$$\int \frac{5}{x^{66}}\, dx = - \frac{1}{13 x^{65}} + C$$$A