Integralen av $$$3^{2 x}$$$

Bestäm $$$\int 3^{2 x}\, dx$$$.

Inmatningen skrivs om: $$$\int{3^{2 x} d x}=\int{9^{x} d x}$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=9$$$:

$${\color{red}{\int{9^{x} d x}}} = {\color{red}{\frac{9^{x}}{\ln{\left(9 \right)}}}}$$

Alltså,

$$\int{9^{x} d x} = \frac{9^{x}}{\ln{\left(9 \right)}}$$

Förenkla:

$$\int{9^{x} d x} = \frac{9^{x}}{2 \ln{\left(3 \right)}}$$

Lägg till integrationskonstanten:

$$\int{9^{x} d x} = \frac{9^{x}}{2 \ln{\left(3 \right)}}+C$$

$$$\int 3^{2 x}\, dx = \frac{9^{x}}{2 \ln\left(3\right)} + C$$$A