Calculus Examples

$\frac{x}{2} d x$

Step 1

Write $\frac{x}{2} d x$ as a function.

$f (x) = \frac{x}{2} d x$

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.

$F (x) = \int \frac{x}{2} \cdot (d x) d x$

Step 4

Simplify.

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Step 4.1

Combine $d$ and $\frac{x}{2}$ .

$\int \frac{d x}{2} \cdot x d x$

Step 4.2

Combine $\frac{d x}{2}$ and $x$ .

$\int \frac{d x \cdot x}{2} d x$

Step 4.3

Raise $x$ to the power of $1$ .

$\int \frac{d (x^{1} x)}{2} d x$

Step 4.4

Raise $x$ to the power of $1$ .

$\int \frac{d (x^{1} x^{1})}{2} d x$

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

$\int \frac{d x^{2}}{2} d x$

Step 5

Since $\frac{d}{2}$ is constant with respect to $x$ , move $\frac{d}{2}$ out of the integral.

$\frac{d}{2} \int x^{2} d x$

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

Combine $\frac{1}{2}$ and $d$ .

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

Step 7.3.2

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

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

Step 7.3.3

Multiply $2$ by $3$ .

$\frac{d}{6} x^{3} + C$

Step 7.3.4

Combine $\frac{d}{6}$ and $x^{3}$ .

$\frac{d x^{3}}{6} + C$

$\frac{1}{6} d x^{3} + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{x}{2} \cdot (d x)$ .

$F (x) =$ $\frac{1}{6} d x^{3} + C$