Find the Function f(x)=1/3*4^x

$f (x) = \frac{1}{3} \cdot 4^{x}$

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int 4^{x} d x$

The integral of $4^{x}$ with respect to $x$ is $\frac{4^{x}}{\ln (4)}$ .

$\frac{1}{3} (\frac{4^{x}}{\ln (4)} + C)$

Simplify the answer.

Rewrite $\frac{1}{3} (\frac{4^{x}}{\ln (4)} + C)$ as $\frac{1}{3} \cdot \frac{1}{\ln (4)} \cdot 4^{x} + C$ .

$\frac{1}{3} \cdot \frac{1}{\ln (4)} \cdot 4^{x} + C$

Multiply $\frac{1}{3}$ by $\frac{1}{\ln (4)}$ .

$\frac{1}{3 \ln (4)} \cdot 4^{x} + C$

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$F (x) = \frac{1}{3 \ln (4)} \cdot 4^{x} + C$