Integralen av $$$2 x e^{2} - 4$$$

Bestäm $$$\int \left(2 x e^{2} - 4\right)\, dx$$$.

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=4$$$:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2 e^{2}$$$ och $$$f{\left(x \right)} = x$$$:

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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