Calculus Examples

$\frac{d y}{d x} = \frac{2 x y}{(x^{2} - 2) (y^{2} + 3)}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{2 x}{x^{2} - 2} \cdot \frac{y}{y^{2} + 3}$

Step 1.2

Multiply both sides by $\frac{y^{2} + 3}{y}$ .

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{y^{2} + 3}{y} (\frac{2 x}{x^{2} - 2} \cdot \frac{y}{y^{2} + 3})$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Multiply $\frac{2 x}{x^{2} - 2}$ by $\frac{y}{y^{2} + 3}$ .

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{y^{2} + 3}{y} \cdot \frac{2 x y}{(x^{2} - 2) (y^{2} + 3)}$

Step 1.3.2

Cancel the common factor of $y^{2} + 3$ .

Tap for more steps...

Step 1.3.2.1

Factor $y^{2} + 3$ out of $(x^{2} - 2) (y^{2} + 3)$ .

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{y^{2} + 3}{y} \cdot \frac{2 x y}{(y^{2} + 3) (x^{2} - 2)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{y^{2} + 3}{y} \cdot \frac{2 x y}{(y^{2} + 3) (x^{2} - 2)}$

Step 1.3.2.3

Rewrite the expression.

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{2 x y}{x^{2} - 2}$

Step 1.3.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.3.3.1

Factor $y$ out of $2 x y$ .

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y (2 x)}{x^{2} - 2}$

Step 1.3.3.2

Cancel the common factor.

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{1}{y} \cdot \frac{y (2 x)}{x^{2} - 2}$

Step 1.3.3.3

Rewrite the expression.

$\frac{y^{2} + 3}{y} \frac{d y}{d x} = \frac{2 x}{x^{2} - 2}$

Step 1.4

Rewrite the equation.

$\frac{y^{2} + 3}{y} d y = \frac{2 x}{x^{2} - 2} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{y^{2} + 3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Split the fraction into multiple fractions.

$\int \frac{y^{2}}{y} + \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{y^{2}}{y} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3

Cancel the common factor of $y^{2}$ and $y$ .

Tap for more steps...

Step 2.2.3.1

Factor $y$ out of $y^{2}$ .

$\int \frac{y \cdot y}{y} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.2.3.2.1

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

$\int \frac{y \cdot y}{y^{1}} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3.2.2

Factor $y$ out of $y^{1}$ .

$\int \frac{y \cdot y}{y \cdot 1} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3.2.3

Cancel the common factor.

$\int \frac{y \cdot y}{y \cdot 1} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3.2.4

Rewrite the expression.

$\int \frac{y}{1} d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.3.2.5

Divide $y$ by $1$ .

$\int y d y + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.4

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

$\frac{1}{2} y^{2} + C_{1} + \int \frac{3}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.5

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} + 3 \int \frac{1}{y} d y = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.6

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\frac{1}{2} y^{2} + C_{1} + 3 (\ln (| y |) + C_{2}) = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.2.7

Simplify.

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \int \frac{2 x}{x^{2} - 2} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 2 \int \frac{x}{x^{2} - 2} d x$

Step 2.3.2

Let $u = x^{2} - 2$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.3.2.1

Let $u = x^{2} - 2$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.3.2.1.1

Differentiate $x^{2} - 2$ .

$\frac{d}{d x} [x^{2} - 2]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2]$

Step 2.3.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 2]$

Step 2.3.2.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.3.2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 2 \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 2.3.3

Simplify.

Tap for more steps...

Step 2.3.3.1

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

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 2 \int \frac{1}{u \cdot 2} d u$

Step 2.3.3.2

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

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 2 \int \frac{1}{2 u} d u$

Step 2.3.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 2 (\frac{1}{2} \int \frac{1}{u} d u)$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \frac{2}{2} \int \frac{1}{u} d u$

Step 2.3.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.5.2.1

Cancel the common factor.

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \frac{2}{2} \int \frac{1}{u} d u$

Step 2.3.5.2.2

Rewrite the expression.

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = 1 \int \frac{1}{u} d u$

Step 2.3.5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \int \frac{1}{u} d u$

Step 2.3.6

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \ln (| u |) + C_{4}$

Step 2.3.7

Replace all occurrences of $u$ with $x^{2} - 2$ .

$\frac{1}{2} y^{2} + 3 \ln (| y |) + C_{3} = \ln (| x^{2} - 2 |) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\frac{1}{2} y^{2} + 3 \ln (| y |) = \ln (| x^{2} - 2 |) + C$