Integralen av $$$\frac{2^{a}}{b}$$$ med avseende på $$$a$$$

Bestäm $$$\int \frac{2^{a}}{b}\, da$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ med $$$c=\frac{1}{b}$$$ och $$$f{\left(a \right)} = 2^{a}$$$:

$${\color{red}{\int{\frac{2^{a}}{b} d a}}} = {\color{red}{\frac{\int{2^{a} d a}}{b}}}$$

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

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

Alltså,

$$\int{\frac{2^{a}}{b} d a} = \frac{2^{a}}{b \ln{\left(2 \right)}}$$

Lägg till integrationskonstanten:

$$\int{\frac{2^{a}}{b} d a} = \frac{2^{a}}{b \ln{\left(2 \right)}}+C$$

$$$\int \frac{2^{a}}{b}\, da = \frac{2^{a}}{b \ln\left(2\right)} + C$$$A