Integralen av $$$\frac{1}{x^{415}}$$$

Bestäm $$$\int \frac{1}{x^{415}}\, dx$$$.

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-415$$$:

$${\color{red}{\int{\frac{1}{x^{415}} d x}}}={\color{red}{\int{x^{-415} d x}}}={\color{red}{\frac{x^{-415 + 1}}{-415 + 1}}}={\color{red}{\left(- \frac{x^{-414}}{414}\right)}}={\color{red}{\left(- \frac{1}{414 x^{414}}\right)}}$$

Alltså,

$$\int{\frac{1}{x^{415}} d x} = - \frac{1}{414 x^{414}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{x^{415}} d x} = - \frac{1}{414 x^{414}}+C$$

$$$\int \frac{1}{x^{415}}\, dx = - \frac{1}{414 x^{414}} + C$$$A