Integralen av $$$- 2 y^{58} - 12$$$

Bestäm $$$\int \left(- 2 y^{58} - 12\right)\, dy$$$.

Integrera termvis:

$${\color{red}{\int{\left(- 2 y^{58} - 12\right)d y}}} = {\color{red}{\left(- \int{12 d y} - \int{2 y^{58} d y}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dy = c y$$$ med $$$c=12$$$:

$$- \int{2 y^{58} d y} - {\color{red}{\int{12 d y}}} = - \int{2 y^{58} d y} - {\color{red}{\left(12 y\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ med $$$c=2$$$ och $$$f{\left(y \right)} = y^{58}$$$:

$$- 12 y - {\color{red}{\int{2 y^{58} d y}}} = - 12 y - {\color{red}{\left(2 \int{y^{58} d y}\right)}}$$

Tillämpa potensregeln $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=58$$$:

$$- 12 y - 2 {\color{red}{\int{y^{58} d y}}}=- 12 y - 2 {\color{red}{\frac{y^{1 + 58}}{1 + 58}}}=- 12 y - 2 {\color{red}{\left(\frac{y^{59}}{59}\right)}}$$

Alltså,

$$\int{\left(- 2 y^{58} - 12\right)d y} = - \frac{2 y^{59}}{59} - 12 y$$

Förenkla:

$$\int{\left(- 2 y^{58} - 12\right)d y} = \frac{2 y \left(- y^{58} - 354\right)}{59}$$

Lägg till integrationskonstanten:

$$\int{\left(- 2 y^{58} - 12\right)d y} = \frac{2 y \left(- y^{58} - 354\right)}{59}+C$$

$$$\int \left(- 2 y^{58} - 12\right)\, dy = \frac{2 y \left(- y^{58} - 354\right)}{59} + C$$$A