Calculus Examples

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

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.

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

Step 3

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

$\frac{1}{3} \int x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $\frac{1}{3} (\frac{1}{2} x^{2} + C)$ as $\frac{1}{3} \cdot \frac{1}{2} x^{2} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

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

Step 5.2.2

Multiply $3$ by $2$ .

$\frac{1}{6} x^{2} + C$

Step 6

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

$F (x) =$ $\frac{1}{6} x^{2} + C$

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