Integralen av $$$\frac{x^{2}}{49}$$$

Bestäm $$$\int \frac{x^{2}}{49}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{49}$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\frac{x^{2}}{49} d x}}} = {\color{red}{\left(\frac{\int{x^{2} d x}}{49}\right)}}$$

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

$$\frac{{\color{red}{\int{x^{2} d x}}}}{49}=\frac{{\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{49}=\frac{{\color{red}{\left(\frac{x^{3}}{3}\right)}}}{49}$$

Alltså,

$$\int{\frac{x^{2}}{49} d x} = \frac{x^{3}}{147}$$

Lägg till integrationskonstanten:

$$\int{\frac{x^{2}}{49} d x} = \frac{x^{3}}{147}+C$$

$$$\int \frac{x^{2}}{49}\, dx = \frac{x^{3}}{147} + C$$$A