Integralen av $$$- x^{17} + x^{2} - 5 x$$$

Bestäm $$$\int \left(- x^{17} + x^{2} - 5 x\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(- x^{17} + x^{2} - 5 x\right)d x}}} = {\color{red}{\left(- \int{5 x d x} + \int{x^{2} d x} - \int{x^{17} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

$$- \int{5 x d x} - \int{x^{17} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{5 x d x} - \int{x^{17} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{5 x d x} - \int{x^{17} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

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

$$\frac{x^{3}}{3} - \int{5 x d x} - {\color{red}{\int{x^{17} d x}}}=\frac{x^{3}}{3} - \int{5 x d x} - {\color{red}{\frac{x^{1 + 17}}{1 + 17}}}=\frac{x^{3}}{3} - \int{5 x d x} - {\color{red}{\left(\frac{x^{18}}{18}\right)}}$$

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

$$- \frac{x^{18}}{18} + \frac{x^{3}}{3} - {\color{red}{\int{5 x d x}}} = - \frac{x^{18}}{18} + \frac{x^{3}}{3} - {\color{red}{\left(5 \int{x d x}\right)}}$$

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

$$- \frac{x^{18}}{18} + \frac{x^{3}}{3} - 5 {\color{red}{\int{x d x}}}=- \frac{x^{18}}{18} + \frac{x^{3}}{3} - 5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \frac{x^{18}}{18} + \frac{x^{3}}{3} - 5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(- x^{17} + x^{2} - 5 x\right)d x} = - \frac{x^{18}}{18} + \frac{x^{3}}{3} - \frac{5 x^{2}}{2}$$

Förenkla:

$$\int{\left(- x^{17} + x^{2} - 5 x\right)d x} = \frac{x^{2} \left(- x^{16} + 6 x - 45\right)}{18}$$

Lägg till integrationskonstanten:

$$\int{\left(- x^{17} + x^{2} - 5 x\right)d x} = \frac{x^{2} \left(- x^{16} + 6 x - 45\right)}{18}+C$$

$$$\int \left(- x^{17} + x^{2} - 5 x\right)\, dx = \frac{x^{2} \left(- x^{16} + 6 x - 45\right)}{18} + C$$$A