Calculus Examples

$\frac{x}{y}$

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

Move $2$ to the left of $y$ .

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

Step 6.2.3

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

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

Step 7

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

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