Integralen av $$$20 - \frac{5 x}{2}$$$

Bestäm $$$\int \left(20 - \frac{5 x}{2}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(20 - \frac{5 x}{2}\right)d x}}} = {\color{red}{\left(\int{20 d x} - \int{\frac{5 x}{2} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=20$$$:

$$- \int{\frac{5 x}{2} d x} + {\color{red}{\int{20 d x}}} = - \int{\frac{5 x}{2} d x} + {\color{red}{\left(20 x\right)}}$$

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

$$20 x - {\color{red}{\int{\frac{5 x}{2} d x}}} = 20 x - {\color{red}{\left(\frac{5 \int{x d x}}{2}\right)}}$$

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

$$20 x - \frac{5 {\color{red}{\int{x d x}}}}{2}=20 x - \frac{5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{2}=20 x - \frac{5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{2}$$

Alltså,

$$\int{\left(20 - \frac{5 x}{2}\right)d x} = - \frac{5 x^{2}}{4} + 20 x$$

Förenkla:

$$\int{\left(20 - \frac{5 x}{2}\right)d x} = \frac{5 x \left(16 - x\right)}{4}$$

Lägg till integrationskonstanten:

$$\int{\left(20 - \frac{5 x}{2}\right)d x} = \frac{5 x \left(16 - x\right)}{4}+C$$

$$$\int \left(20 - \frac{5 x}{2}\right)\, dx = \frac{5 x \left(16 - x\right)}{4} + C$$$A