Find the Function f(x)=x^2+2x+1

$f (x) = x^{2} + 2 x + 1$

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.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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

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

Simplify.

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

Reorder terms.

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

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.

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