Integralen av $$$\frac{w^{2}}{2 e^{6}}$$$

Bestäm $$$\int \frac{w^{2}}{2 e^{6}}\, dw$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ med $$$c=\frac{1}{2 e^{6}}$$$ och $$$f{\left(w \right)} = w^{2}$$$:

$${\color{red}{\int{\frac{w^{2}}{2 e^{6}} d w}}} = {\color{red}{\left(\frac{\int{w^{2} d w}}{2 e^{6}}\right)}}$$

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

$$\frac{{\color{red}{\int{w^{2} d w}}}}{2 e^{6}}=\frac{{\color{red}{\frac{w^{1 + 2}}{1 + 2}}}}{2 e^{6}}=\frac{{\color{red}{\left(\frac{w^{3}}{3}\right)}}}{2 e^{6}}$$

Alltså,

$$\int{\frac{w^{2}}{2 e^{6}} d w} = \frac{w^{3}}{6 e^{6}}$$

Lägg till integrationskonstanten:

$$\int{\frac{w^{2}}{2 e^{6}} d w} = \frac{w^{3}}{6 e^{6}}+C$$

$$$\int \frac{w^{2}}{2 e^{6}}\, dw = \frac{w^{3}}{6 e^{6}} + C$$$A