Find the Function f(x)=1/4*2^x

$f (x) = \frac{1}{4} \cdot 2^{x}$

Step 1

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

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

Step 2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int 2^{x} d x$

Step 3

The integral of $2^{x}$ with respect to $x$ is $\frac{2^{x}}{\ln (2)}$ .

$\frac{1}{4} (\frac{2^{x}}{\ln (2)} + C)$

Step 4

Simplify the answer.

Rewrite $\frac{1}{4} (\frac{2^{x}}{\ln (2)} + C)$ as $\frac{1}{4} \cdot \frac{1}{\ln (2)} \cdot 2^{x} + C$ .

$\frac{1}{4} \cdot \frac{1}{\ln (2)} \cdot 2^{x} + C$

Multiply $\frac{1}{4}$ by $\frac{1}{\ln (2)}$ .

$\frac{1}{4 \ln (2)} \cdot 2^{x} + C$

Step 5

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

$F (x) = \frac{1}{4 \ln (2)} \cdot 2^{x} + C$