Integralen av $$$\frac{4 x}{23}$$$

Bestäm $$$\int \frac{4 x}{23}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{4}{23}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{4 x}{23} d x}}} = {\color{red}{\left(\frac{4 \int{x d x}}{23}\right)}}$$

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

$$\frac{4 {\color{red}{\int{x d x}}}}{23}=\frac{4 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{23}=\frac{4 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{23}$$

Alltså,

$$\int{\frac{4 x}{23} d x} = \frac{2 x^{2}}{23}$$

Lägg till integrationskonstanten:

$$\int{\frac{4 x}{23} d x} = \frac{2 x^{2}}{23}+C$$

$$$\int \frac{4 x}{23}\, dx = \frac{2 x^{2}}{23} + C$$$A