Integralen av $$$y^{4} - y$$$

Bestäm $$$\int \left(y^{4} - y\right)\, dy$$$.

Integrera termvis:

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

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

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

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

$$\frac{y^{5}}{5} - {\color{red}{\int{y d y}}}=\frac{y^{5}}{5} - {\color{red}{\frac{y^{1 + 1}}{1 + 1}}}=\frac{y^{5}}{5} - {\color{red}{\left(\frac{y^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(y^{4} - y\right)d y} = \frac{y^{5}}{5} - \frac{y^{2}}{2}$$

Lägg till integrationskonstanten:

$$\int{\left(y^{4} - y\right)d y} = \frac{y^{5}}{5} - \frac{y^{2}}{2}+C$$

$$$\int \left(y^{4} - y\right)\, dy = \left(\frac{y^{5}}{5} - \frac{y^{2}}{2}\right) + C$$$A