Integralen av $$$2 x^{\frac{4}{3}}$$$

Bestäm $$$\int 2 x^{\frac{4}{3}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = x^{\frac{4}{3}}$$$:

$${\color{red}{\int{2 x^{\frac{4}{3}} d x}}} = {\color{red}{\left(2 \int{x^{\frac{4}{3}} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{4}{3}$$$:

$$2 {\color{red}{\int{x^{\frac{4}{3}} d x}}}=2 {\color{red}{\frac{x^{1 + \frac{4}{3}}}{1 + \frac{4}{3}}}}=2 {\color{red}{\left(\frac{3 x^{\frac{7}{3}}}{7}\right)}}$$

Alltså,

$$\int{2 x^{\frac{4}{3}} d x} = \frac{6 x^{\frac{7}{3}}}{7}$$

Lägg till integrationskonstanten:

$$\int{2 x^{\frac{4}{3}} d x} = \frac{6 x^{\frac{7}{3}}}{7}+C$$

$$$\int 2 x^{\frac{4}{3}}\, dx = \frac{6 x^{\frac{7}{3}}}{7} + C$$$A