Calculus Examples

$(\frac{1}{3}) x^{3} - 5 x$

Step 1

Write $(\frac{1}{3}) x^{3} - 5 x$ as a function.

$f (x) = (\frac{1}{3}) x^{3} - 5 x$

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.

$F (x) = \int (\frac{1}{3}) x^{3} - 5 x d x$

Step 4

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

$\int \frac{x^{3}}{3} - 5 x d x$

Step 5

Split the single integral into multiple integrals.

$\int \frac{x^{3}}{3} d x + \int - 5 x d x$

Step 6

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

$\frac{1}{3} \int x^{3} d x + \int - 5 x d x$

Step 7

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

$\frac{1}{3} (\frac{1}{4} x^{4} + C) + \int - 5 x d x$

Step 8

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{1}{3} (\frac{1}{4} x^{4} + C) - 5 \int x d x$

Step 9

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

$\frac{1}{3} (\frac{1}{4} x^{4} + C) - 5 (\frac{1}{2} x^{2} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{3} \cdot \frac{1}{4} x^{4} - 5 (\frac{1}{2}) x^{2} + C$

Step 10.2

Simplify.

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Step 10.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{4}$ .

$\frac{1}{3 \cdot 4} x^{4} - 5 (\frac{1}{2}) x^{2} + C$

Step 10.2.2

Multiply $3$ by $4$ .

$\frac{1}{12} x^{4} - 5 (\frac{1}{2}) x^{2} + C$

Step 10.2.3

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{1}{12} x^{4} + \frac{- 5}{2} x^{2} + C$

Step 10.2.4

Move the negative in front of the fraction.

$\frac{1}{12} x^{4} - \frac{5}{2} x^{2} + C$

Step 11

The answer is the antiderivative of the function $f (x) = (\frac{1}{3}) x^{3} - 5 x$ .

$F (x) =$ $\frac{1}{12} x^{4} - \frac{5}{2} x^{2} + C$