Find the Function f(x)=(x+3)/2

$f (x) = \frac{x + 3}{2}$

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

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

$\frac{1}{2} \int x + 3 d x$

Step 3

Split the single integral into multiple integrals.

$\frac{1}{2} (\int x d x + \int 3 d x)$

Step 4

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C + \int 3 d x)$

Step 5

Apply the constant rule.

$\frac{1}{2} (\frac{1}{2} x^{2} + C + 3 x + C)$

Step 6

Simplify.

$\frac{1}{2} (\frac{1}{2} x^{2} + 3 x) + C$

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.

$F (x) = \frac{1}{2} \cdot (\frac{1}{2} x^{2} + 3 x) + C$