Integralen av $$$\frac{x}{2} + 1$$$

Bestäm $$$\int \left(\frac{x}{2} + 1\right)\, dx$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\int{\frac{x}{2} d x} + {\color{red}{\int{1 d x}}} = \int{\frac{x}{2} d x} + {\color{red}{x}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = x$$$:

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

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

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

Alltså,

$$\int{\left(\frac{x}{2} + 1\right)d x} = \frac{x^{2}}{4} + x$$

Förenkla:

$$\int{\left(\frac{x}{2} + 1\right)d x} = \frac{x \left(x + 4\right)}{4}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{x}{2} + 1\right)d x} = \frac{x \left(x + 4\right)}{4}+C$$

$$$\int \left(\frac{x}{2} + 1\right)\, dx = \frac{x \left(x + 4\right)}{4} + C$$$A