Integralen av $$$3 x^{4} - 1$$$

Bestäm $$$\int \left(3 x^{4} - 1\right)\, dx$$$.

Integrera termvis:

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

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

$$\int{3 x^{4} d x} - {\color{red}{\int{1 d x}}} = \int{3 x^{4} d x} - {\color{red}{x}}$$

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

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

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

$$- x + 3 {\color{red}{\int{x^{4} d x}}}=- x + 3 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- x + 3 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Alltså,

$$\int{\left(3 x^{4} - 1\right)d x} = \frac{3 x^{5}}{5} - x$$

Lägg till integrationskonstanten:

$$\int{\left(3 x^{4} - 1\right)d x} = \frac{3 x^{5}}{5} - x+C$$

$$$\int \left(3 x^{4} - 1\right)\, dx = \left(\frac{3 x^{5}}{5} - x\right) + C$$$A