Find the Function f(x)=X^2-4x+4

$f (x) = X^{2} - 4 x + 4$

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.

$\int X^{2} d x + \int - 4 x d x + \int 4 d x$

Step 3

Apply the constant rule.

$X^{2} x + C + \int - 4 x d x + \int 4 d x$

Step 4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$X^{2} x + C - 4 \int x d x + \int 4 d x$

Step 5

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

$X^{2} x + C - 4 (\frac{1}{2} x^{2} + C) + \int 4 d x$

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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

$X^{2} x + C - 4 (\frac{x^{2}}{2} + C) + 4 x + C$

Simplify.

$X^{2} x - 2 x^{2} + 4 x + C$

Step 8

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

$F (x) = X^{2} x - 2 x^{2} + 4 x + C$