Calculus Examples

$f' (x) = \frac{x^{3}}{3} - x^{2} - 3 x$

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.

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{x^{4}}{12} - \frac{x^{3}}{3} - 3 (\frac{1}{2} x^{2}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{4}}{12} - \frac{x^{3}}{3} - 3 \frac{x^{2}}{2} + C$

Step 9.2.2

Combine $- 3$ and $\frac{x^{2}}{2}$ .

$\frac{x^{4}}{12} - \frac{x^{3}}{3} + \frac{- 3 x^{2}}{2} + C$

Step 9.2.3

Move the negative in front of the fraction.

$\frac{x^{4}}{12} - \frac{x^{3}}{3} - \frac{3 x^{2}}{2} + C$

Step 10

Reorder terms.

$\frac{1}{12} x^{4} - \frac{1}{3} x^{3} - \frac{3}{2} x^{2} + C$

Step 11

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

$f (x) = \frac{1}{12} x^{4} - \frac{1}{3} x^{3} - \frac{3}{2} x^{2} + C$