Integralen av $$$\frac{x^{5} - 4}{x^{22}}$$$

Bestäm $$$\int \frac{x^{5} - 4}{x^{22}}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{x^{5} - 4}{x^{22}} d x}}} = {\color{red}{\int{\left(\frac{1}{x^{17}} - \frac{4}{x^{22}}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(\frac{1}{x^{17}} - \frac{4}{x^{22}}\right)d x}}} = {\color{red}{\left(- \int{\frac{4}{x^{22}} d x} + \int{\frac{1}{x^{17}} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-17$$$:

$$- \int{\frac{4}{x^{22}} d x} + {\color{red}{\int{\frac{1}{x^{17}} d x}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\int{x^{-17} d x}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\frac{x^{-17 + 1}}{-17 + 1}}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\left(- \frac{x^{-16}}{16}\right)}}=- \int{\frac{4}{x^{22}} d x} + {\color{red}{\left(- \frac{1}{16 x^{16}}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=4$$$ och $$$f{\left(x \right)} = \frac{1}{x^{22}}$$$:

$$- {\color{red}{\int{\frac{4}{x^{22}} d x}}} - \frac{1}{16 x^{16}} = - {\color{red}{\left(4 \int{\frac{1}{x^{22}} d x}\right)}} - \frac{1}{16 x^{16}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-22$$$:

$$- 4 {\color{red}{\int{\frac{1}{x^{22}} d x}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\int{x^{-22} d x}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\frac{x^{-22 + 1}}{-22 + 1}}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\left(- \frac{x^{-21}}{21}\right)}} - \frac{1}{16 x^{16}}=- 4 {\color{red}{\left(- \frac{1}{21 x^{21}}\right)}} - \frac{1}{16 x^{16}}$$

Alltså,

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = - \frac{1}{16 x^{16}} + \frac{4}{21 x^{21}}$$

Förenkla:

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = \frac{64 - 21 x^{5}}{336 x^{21}}$$

Lägg till integrationskonstanten:

$$\int{\frac{x^{5} - 4}{x^{22}} d x} = \frac{64 - 21 x^{5}}{336 x^{21}}+C$$

$$$\int \frac{x^{5} - 4}{x^{22}}\, dx = \frac{64 - 21 x^{5}}{336 x^{21}} + C$$$A