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

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

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.

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

Step 3

Apply the constant rule.

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

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 5

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

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

Step 6

Simplify.

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

Step 7

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

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