Integralen av $$$\left(\frac{11}{5}\right)^{x}$$$

Bestäm $$$\int \left(\frac{11}{5}\right)^{x}\, dx$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{11}{5}$$$:

$${\color{red}{\int{\left(\frac{11}{5}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{11}{5}\right)^{x}}{\ln{\left(\frac{11}{5} \right)}}}}$$

Alltså,

$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{\ln{\left(\frac{11}{5} \right)}}$$

Förenkla:

$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln{\left(5 \right)} + \ln{\left(11 \right)}}$$

Lägg till integrationskonstanten:

$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln{\left(5 \right)} + \ln{\left(11 \right)}}+C$$

$$$\int \left(\frac{11}{5}\right)^{x}\, dx = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln\left(5\right) + \ln\left(11\right)} + C$$$A