Integralen av $$$- 5 x^{9} + 3 x^{5}$$$

Bestäm $$$\int \left(- 5 x^{9} + 3 x^{5}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(- 5 x^{9} + 3 x^{5}\right)d x}}} = {\color{red}{\left(\int{3 x^{5} d x} - \int{5 x^{9} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = x^{9}$$$:

$$\int{3 x^{5} d x} - {\color{red}{\int{5 x^{9} d x}}} = \int{3 x^{5} d x} - {\color{red}{\left(5 \int{x^{9} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=9$$$:

$$\int{3 x^{5} d x} - 5 {\color{red}{\int{x^{9} d x}}}=\int{3 x^{5} d x} - 5 {\color{red}{\frac{x^{1 + 9}}{1 + 9}}}=\int{3 x^{5} d x} - 5 {\color{red}{\left(\frac{x^{10}}{10}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=3$$$ och $$$f{\left(x \right)} = x^{5}$$$:

$$- \frac{x^{10}}{2} + {\color{red}{\int{3 x^{5} d x}}} = - \frac{x^{10}}{2} + {\color{red}{\left(3 \int{x^{5} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=5$$$:

$$- \frac{x^{10}}{2} + 3 {\color{red}{\int{x^{5} d x}}}=- \frac{x^{10}}{2} + 3 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=- \frac{x^{10}}{2} + 3 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Alltså,

$$\int{\left(- 5 x^{9} + 3 x^{5}\right)d x} = - \frac{x^{10}}{2} + \frac{x^{6}}{2}$$

Förenkla:

$$\int{\left(- 5 x^{9} + 3 x^{5}\right)d x} = \frac{x^{6} \left(1 - x^{4}\right)}{2}$$

Lägg till integrationskonstanten:

$$\int{\left(- 5 x^{9} + 3 x^{5}\right)d x} = \frac{x^{6} \left(1 - x^{4}\right)}{2}+C$$

$$$\int \left(- 5 x^{9} + 3 x^{5}\right)\, dx = \frac{x^{6} \left(1 - x^{4}\right)}{2} + C$$$A