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

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

Integrera termvis:

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

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

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

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

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

$$\int{\left(4 x^{3} - x\right)d x} = x^{4} - \frac{x^{2}}{2}+C$$

$$$\int \left(4 x^{3} - x\right)\, dx = \left(x^{4} - \frac{x^{2}}{2}\right) + C$$$A