Integralen av $$$\left(\frac{e}{2}\right)^{x}$$$

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

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{e}{2}$$$:

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

Alltså,

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

Förenkla:

$$\int{\left(\frac{e}{2}\right)^{x} d x} = \frac{\left(\frac{e}{2}\right)^{x}}{1 - \ln{\left(2 \right)}}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{e}{2}\right)^{x} d x} = \frac{\left(\frac{e}{2}\right)^{x}}{1 - \ln{\left(2 \right)}}+C$$

$$$\int \left(\frac{e}{2}\right)^{x}\, dx = \frac{\left(\frac{e}{2}\right)^{x}}{1 - \ln\left(2\right)} + C$$$A