Calculus Examples

$f (x) = x - 9 x^{2} + 4$

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.

$F (x) = \int x - 9 x^{2} + 4 d x$

Step 3

Split the single integral into multiple integrals.

$\int x d x + \int - 9 x^{2} d x + \int 4 d x$

Step 4

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

$\frac{1}{2} x^{2} + C + \int - 9 x^{2} d x + \int 4 d x$

Step 5

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$\frac{1}{2} x^{2} + C - 9 \int x^{2} d x + \int 4 d x$

Step 6

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

$\frac{1}{2} x^{2} + C - 9 (\frac{1}{3} x^{3} + C) + \int 4 d x$

Step 7

Apply the constant rule.

$\frac{1}{2} x^{2} + C - 9 (\frac{1}{3} x^{3} + C) + 4 x + C$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{2} x^{2} + C - 9 (\frac{x^{3}}{3} + C) + 4 x + C$

Step 8.2

Simplify.

$\frac{x^{2}}{2} - 3 x^{3} + 4 x + C$

Step 8.3

Reorder terms.

$\frac{1}{2} x^{2} - 3 x^{3} + 4 x + C$

Step 9

The answer is the antiderivative of the function $f (x) = x - 9 x^{2} + 4$ .

$F (x) =$ $\frac{1}{2} x^{2} - 3 x^{3} + 4 x + C$