Find the Function f(x)=ax^2+bx+c

$f (x) = a x^{2} + b x + c$

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.

$\int a x^{2} d x + \int b x d x + \int c d x$

Step 3

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

$a \int x^{2} d x + \int b x d x + \int c d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$a (\frac{1}{3} x^{3} + C) + \int b x d x + \int c d x$

Step 5

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

$a (\frac{1}{3} x^{3} + C) + b \int x d x + \int c d x$

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

Combine $\frac{1}{3}$ and $x^{3}$ .

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

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

$a (\frac{x^{3}}{3} + C) + b (\frac{x^{2}}{2} + C) + c x + C$

Simplify.

$\frac{1}{3} a x^{3} + \frac{1}{2} b x^{2} + c x + C$

Step 9

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

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