Find the Function f(x)=ax+b

$f (x) = a x + b$

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

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

Split the single integral into multiple integrals.

$\int a x d x + \int b d x$

Since $a$ is constant with respect to $x$ , move $a$ out of the integral.

$a \int x d x + \int b d x$

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

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

Apply the constant rule.

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

Simplify.

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Simplify.

$\frac{1}{2} a x^{2} + b x + C$

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

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