Integralen av $$$3 x^{2} + x - 1$$$

Bestäm $$$\int \left(3 x^{2} + x - 1\right)\, dx$$$.

Integrera termvis:

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

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

$$\int{x d x} + \int{3 x^{2} d x} - {\color{red}{\int{1 d x}}} = \int{x d x} + \int{3 x^{2} d x} - {\color{red}{x}}$$

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

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

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

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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