Integralen av $$$x^{2} + 1$$$

Bestäm $$$\int \left(x^{2} + 1\right)\, dx$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\int{x^{2} d x} + {\color{red}{\int{1 d x}}} = \int{x^{2} d x} + {\color{red}{x}}$$

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

$$$\int \left(x^{2} + 1\right)\, dx = \left(\frac{x^{3}}{3} + x\right) + C$$$A