Integralen av $$$x^{\frac{5}{2}} - 3$$$

Bestäm $$$\int \left(x^{\frac{5}{2}} - 3\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{\frac{5}{2}} - 3\right)d x}}} = {\color{red}{\left(- \int{3 d x} + \int{x^{\frac{5}{2}} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=3$$$:

$$\int{x^{\frac{5}{2}} d x} - {\color{red}{\int{3 d x}}} = \int{x^{\frac{5}{2}} d x} - {\color{red}{\left(3 x\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=\frac{5}{2}$$$:

$$- 3 x + {\color{red}{\int{x^{\frac{5}{2}} d x}}}=- 3 x + {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=- 3 x + {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

Alltså,

$$\int{\left(x^{\frac{5}{2}} - 3\right)d x} = \frac{2 x^{\frac{7}{2}}}{7} - 3 x$$

Lägg till integrationskonstanten:

$$\int{\left(x^{\frac{5}{2}} - 3\right)d x} = \frac{2 x^{\frac{7}{2}}}{7} - 3 x+C$$

$$$\int \left(x^{\frac{5}{2}} - 3\right)\, dx = \left(\frac{2 x^{\frac{7}{2}}}{7} - 3 x\right) + C$$$A