Calculus Examples

$(1 - x^{2})$

Step 1

Write $(1 - x^{2})$ as a function.

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

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int 1 - x^{2} d x$

Step 4

Remove parentheses.

$\int 1 - x^{2} d x$

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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 9

Simplify.

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

Step 10

The answer is the antiderivative of the function $f (x) = 1 - x^{2}$ .

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