Calculus Examples

$\frac{x y}{2}$

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

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

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

Step 6.3.2

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

$\frac{y}{2 \cdot 2} x^{2} + C$

Step 6.3.3

Multiply $2$ by $2$ .

$\frac{y}{4} x^{2} + C$

Step 6.3.4

Combine $\frac{y}{4}$ and $x^{2}$ .

$\frac{y x^{2}}{4} + C$

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

Step 7

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

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