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

$f (x) = 3 x^{2} + 2 x$

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

Split the single integral into multiple integrals.

$\int 3 x^{2} d x + \int 2 x d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{2} d x + \int 2 x d x$

Step 4

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

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

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 6

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

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

Step 7

Simplify.

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

Simplify.

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $x^{3}$ by $1$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $x^{2}$ by $1$ .

$x^{3} + x^{2} + C$

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.

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