Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{2} - 1$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Simplify the denominator.

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Step 1.2.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

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

Step 1.2.2

Multiply $y^{2} - 1$ by $\frac{x^{2} + 1}{(y + 1) (y - 1)}$ .

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

Step 1.2.3

Simplify the numerator.

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Step 1.2.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

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

Step 1.2.4

Cancel the common factor of $y + 1$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Cancel the common factor of $y - 1$ .

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

Divide $x^{2} + 1$ by $1$ .

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

Step 1.3

Rewrite the equation.

$(y^{2} - 1) d y = (x^{2} + 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

$\frac{1}{3} y^{3} - y + C_{3} = \frac{1}{3} x^{3} + C_{4} + \int d x$

Step 2.3.3

Apply the constant rule.

$\frac{1}{3} y^{3} - y + C_{3} = \frac{1}{3} x^{3} + C_{4} + x + C_{5}$

Step 2.3.4

Simplify.

$\frac{1}{3} y^{3} - y + C_{3} = \frac{1}{3} x^{3} + x + C_{6}$

Step 2.4

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

$\frac{1}{3} y^{3} - y = \frac{1}{3} x^{3} + x + K$