Find the Function f(x)=1/2x+2

$f (x) = \frac{1}{2} x + 2$

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

Split the single integral into multiple integrals.

$\int \frac{1}{2} x d x + \int 2 d x$

Step 3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int x d x + \int 2 d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) + \int 2 d x$

Step 5

Apply the constant rule.

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

Step 6

Simplify.

$\frac{1}{2} \cdot \frac{1}{2} x^{2} + 2 x + C$

Simplify.

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} x^{2} + 2 x + C$

Multiply $2$ by $2$ .

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

Step 7

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

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