Integralen av $$$\frac{x}{8} - 5$$$

Bestäm $$$\int \left(\frac{x}{8} - 5\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(\frac{x}{8} - 5\right)d x}}} = {\color{red}{\left(- \int{5 d x} + \int{\frac{x}{8} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=5$$$:

$$\int{\frac{x}{8} d x} - {\color{red}{\int{5 d x}}} = \int{\frac{x}{8} d x} - {\color{red}{\left(5 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{8}$$$ och $$$f{\left(x \right)} = x$$$:

$$- 5 x + {\color{red}{\int{\frac{x}{8} d x}}} = - 5 x + {\color{red}{\left(\frac{\int{x d x}}{8}\right)}}$$

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

$$- 5 x + \frac{{\color{red}{\int{x d x}}}}{8}=- 5 x + \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{8}=- 5 x + \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{8}$$

Alltså,

$$\int{\left(\frac{x}{8} - 5\right)d x} = \frac{x^{2}}{16} - 5 x$$

Förenkla:

$$\int{\left(\frac{x}{8} - 5\right)d x} = \frac{x \left(x - 80\right)}{16}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{x}{8} - 5\right)d x} = \frac{x \left(x - 80\right)}{16}+C$$

$$$\int \left(\frac{x}{8} - 5\right)\, dx = \frac{x \left(x - 80\right)}{16} + C$$$A