Integralen av $$$x e^{\frac{9}{100}}$$$

Bestäm $$$\int x e^{\frac{9}{100}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=e^{\frac{9}{100}}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{x e^{\frac{9}{100}} d x}}} = {\color{red}{e^{\frac{9}{100}} \int{x d x}}}$$

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

$$e^{\frac{9}{100}} {\color{red}{\int{x d x}}}=e^{\frac{9}{100}} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=e^{\frac{9}{100}} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{x e^{\frac{9}{100}} d x} = \frac{x^{2} e^{\frac{9}{100}}}{2}$$

Lägg till integrationskonstanten:

$$\int{x e^{\frac{9}{100}} d x} = \frac{x^{2} e^{\frac{9}{100}}}{2}+C$$

$$$\int x e^{\frac{9}{100}}\, dx = \frac{x^{2} e^{\frac{9}{100}}}{2} + C$$$A