Integralen av $$$- x + e^{5}$$$

Bestäm $$$\int \left(- x + e^{5}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(- x + e^{5}\right)d x}}} = {\color{red}{\left(- \int{x d x} + \int{e^{5} d x}\right)}}$$

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

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=e^{5}$$$:

$$- \frac{x^{2}}{2} + {\color{red}{\int{e^{5} d x}}} = - \frac{x^{2}}{2} + {\color{red}{x e^{5}}}$$

Alltså,

$$\int{\left(- x + e^{5}\right)d x} = - \frac{x^{2}}{2} + x e^{5}$$

Förenkla:

$$\int{\left(- x + e^{5}\right)d x} = \frac{x \left(- x + 2 e^{5}\right)}{2}$$

Lägg till integrationskonstanten:

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

$$$\int \left(- x + e^{5}\right)\, dx = \frac{x \left(- x + 2 e^{5}\right)}{2} + C$$$A