Integralen av $$$\frac{1}{4 x^{10}}$$$

Bestäm $$$\int \frac{1}{4 x^{10}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{4}$$$ och $$$f{\left(x \right)} = \frac{1}{x^{10}}$$$:

$${\color{red}{\int{\frac{1}{4 x^{10}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{10}} d x}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{\frac{1}{x^{10}} d x}}}}{4}=\frac{{\color{red}{\int{x^{-10} d x}}}}{4}=\frac{{\color{red}{\frac{x^{-10 + 1}}{-10 + 1}}}}{4}=\frac{{\color{red}{\left(- \frac{x^{-9}}{9}\right)}}}{4}=\frac{{\color{red}{\left(- \frac{1}{9 x^{9}}\right)}}}{4}$$

Alltså,

$$\int{\frac{1}{4 x^{10}} d x} = - \frac{1}{36 x^{9}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{4 x^{10}} d x} = - \frac{1}{36 x^{9}}+C$$

$$$\int \frac{1}{4 x^{10}}\, dx = - \frac{1}{36 x^{9}} + C$$$A