Integralen av $$$24 x^{43} e^{2}$$$

Bestäm $$$\int 24 x^{43} e^{2}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=24 e^{2}$$$ och $$$f{\left(x \right)} = x^{43}$$$:

$${\color{red}{\int{24 x^{43} e^{2} d x}}} = {\color{red}{\left(24 e^{2} \int{x^{43} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=43$$$:

$$24 e^{2} {\color{red}{\int{x^{43} d x}}}=24 e^{2} {\color{red}{\frac{x^{1 + 43}}{1 + 43}}}=24 e^{2} {\color{red}{\left(\frac{x^{44}}{44}\right)}}$$

Alltså,

$$\int{24 x^{43} e^{2} d x} = \frac{6 x^{44} e^{2}}{11}$$

Lägg till integrationskonstanten:

$$\int{24 x^{43} e^{2} d x} = \frac{6 x^{44} e^{2}}{11}+C$$

$$$\int 24 x^{43} e^{2}\, dx = \frac{6 x^{44} e^{2}}{11} + C$$$A