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

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

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Split the single integral into multiple integrals.

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

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

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

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

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

Apply the constant rule.

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

Simplify.

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

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

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