Integralen av $$$- x^{2} - \frac{5 x}{2}$$$

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

Integrera termvis:

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

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

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{5}{2}$$$ och $$$f{\left(x \right)} = x$$$:

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

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

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

Alltså,

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

Förenkla:

$$\int{\left(- x^{2} - \frac{5 x}{2}\right)d x} = \frac{x^{2} \left(- 4 x - 15\right)}{12}$$

Lägg till integrationskonstanten:

$$\int{\left(- x^{2} - \frac{5 x}{2}\right)d x} = \frac{x^{2} \left(- 4 x - 15\right)}{12}+C$$

$$$\int \left(- x^{2} - \frac{5 x}{2}\right)\, dx = \frac{x^{2} \left(- 4 x - 15\right)}{12} + C$$$A