Calculus Examples

$f (x) = 100 - x^{2}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 100 - x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 100 d x + \int - x^{2} d x$

Step 4

Apply the constant rule.

$100 x + C + \int - x^{2} d x$

Step 5

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

$100 x + C - \int x^{2} d x$

Step 6

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

$100 x + C - (\frac{1}{3} x^{3} + C)$

Step 7

Simplify.

$100 x - \frac{1}{3} x^{3} + C$

Step 8

The answer is the antiderivative of the function $f (x) = 100 - x^{2}$ .

$F (x) =$ $100 x - \frac{1}{3} x^{3} + C$