Integralen av $$$1 - z^{3}$$$

Bestäm $$$\int \left(1 - z^{3}\right)\, dz$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dz = c z$$$ med $$$c=1$$$:

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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