Find the Function f(x)=6x

$f (x) = 6 x$

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 $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x d x$

Step 3

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

$6 (\frac{1}{2} x^{2} + C)$

Step 4

Simplify the answer.

Rewrite $6 (\frac{1}{2} x^{2} + C)$ as $6 (\frac{1}{2}) x^{2} + C$ .

$6 (\frac{1}{2}) x^{2} + C$

Simplify.

Combine $6$ and $\frac{1}{2}$ .

$\frac{6}{2} x^{2} + C$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2} x^{2} + C$

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

$3 x^{2} + C$

Step 5

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

$F (x) = 3 x^{2} + C$