Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

Factor out the greatest common factor from each group.

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Step 1.1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

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

Step 1.1.2

Factor the polynomial by factoring out the greatest common factor, $1 - x^{2}$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{1}{1 + y^{2}}$ .

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

Step 1.4

Cancel the common factor of $1 + y^{2}$ .

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Step 1.4.1

Factor $1 + y^{2}$ out of $\frac{(1 + x) (1 - x)}{x^{2}} (1 + y^{2})$ .

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

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

Step 1.5

Rewrite the equation.

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

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

The integral of $\frac{1}{1^{2} + y^{2}}$ with respect to $y$ is $\arctan (y) + C_{1}$ .

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\arctan (y) + C_{1} = \int (1 + x) (1 - x) {(x^{2})}^{- 1} d x$

Step 2.3.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arctan (y) + C_{1} = \int (1 + x) (1 - x) x^{2 \cdot - 1} d x$

Step 2.3.2.2

Multiply $2$ by $- 1$ .

$\arctan (y) + C_{1} = \int (1 + x) (1 - x) x^{- 2} d x$

$\arctan (y) + C_{1} = \int (1 + x) (1 - x) x^{- 2} d x$

Step 2.3.3

Expand $(1 + x) (1 - x) x^{- 2}$ .

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Step 2.3.3.1

Apply the distributive property.

$\arctan (y) + C_{1} = \int (1 (1 - x) + x (1 - x)) x^{- 2} d x$

Step 2.3.3.2

Apply the distributive property.

$\arctan (y) + C_{1} = \int (1 \cdot 1 + 1 (- x) + x (1 - x)) x^{- 2} d x$

Step 2.3.3.3

Apply the distributive property.

$\arctan (y) + C_{1} = \int (1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x)) x^{- 2} d x$

Step 2.3.3.4

Apply the distributive property.

$\arctan (y) + C_{1} = \int (1 \cdot 1 + 1 (- x)) x^{- 2} + (x \cdot 1 + x (- x)) x^{- 2} d x$

Step 2.3.3.5

Apply the distributive property.

$\arctan (y) + C_{1} = \int 1 \cdot 1 x^{- 2} + 1 (- x) x^{- 2} + (x \cdot 1 + x (- x)) x^{- 2} d x$

Step 2.3.3.6

Apply the distributive property.

$\arctan (y) + C_{1} = \int 1 \cdot 1 x^{- 2} + 1 (- x) x^{- 2} + x \cdot 1 x^{- 2} + x (- x) x^{- 2} d x$

Step 2.3.3.7

Reorder $x$ and $1$ .

$\arctan (y) + C_{1} = \int 1 \cdot 1 x^{- 2} + 1 \cdot - 1 x \cdot x^{- 2} + 1 \cdot x \cdot x^{- 2} + x (- x) x^{- 2} d x$

Step 2.3.3.8

Reorder $x$ and $- 1$ .

$\arctan (y) + C_{1} = \int 1 \cdot 1 x^{- 2} + 1 \cdot - 1 x \cdot x^{- 2} + 1 \cdot x \cdot x^{- 2} - 1 \cdot x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.9

Multiply $1$ by $1$ .

$\arctan (y) + C_{1} = \int 1 x^{- 2} + 1 \cdot - 1 x \cdot x^{- 2} + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.10

Multiply $x^{- 2}$ by $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} + 1 \cdot - 1 x \cdot x^{- 2} + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.11

Multiply $- 1$ by $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x \cdot x^{- 2} + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.12

Factor out negative.

$\arctan (y) + C_{1} = \int x^{- 2} - (x \cdot x^{- 2}) + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.13

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

$\arctan (y) + C_{1} = \int x^{- 2} - (x^{1} x^{- 2}) + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.14

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{1 - 2} + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.15

Subtract $2$ from $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + 1 x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.16

Multiply $x$ by $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x \cdot x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.17

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{1} x^{- 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.18

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{1 - 2} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.19

Subtract $2$ from $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - 1 x \cdot x \cdot x^{- 2} d x$

Step 2.3.3.20

Factor out negative.

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - (x \cdot x) x^{- 2} d x$

Step 2.3.3.21

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - (x^{1} x) x^{- 2} d x$

Step 2.3.3.22

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - (x^{1} x^{1}) x^{- 2} d x$

Step 2.3.3.23

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - x^{1 + 1} x^{- 2} d x$

Step 2.3.3.24

Add $1$ and $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - x^{2} x^{- 2} d x$

Step 2.3.3.25

Factor out negative.

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - (x^{2} x^{- 2}) d x$

Step 2.3.3.26

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

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - x^{2 - 2} d x$

Step 2.3.3.27

Subtract $2$ from $2$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - x^{0} d x$

Step 2.3.3.28

Anything raised to $0$ is $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - 1 \cdot 1 d x$

Step 2.3.3.29

Multiply $- 1$ by $1$ .

$\arctan (y) + C_{1} = \int x^{- 2} - x^{- 1} + x^{- 1} - 1 d x$

Step 2.3.3.30

Add $- x^{- 1}$ and $x^{- 1}$ .

$\arctan (y) + C_{1} = \int x^{- 2} + 0 - 1 d x$

Step 2.3.3.31

Subtract $1$ from $0$ .

$\arctan (y) + C_{1} = \int x^{- 2} - 1 d x$

$\arctan (y) + C_{1} = \int x^{- 2} - 1 d x$

Step 2.3.4

Split the single integral into multiple integrals.

$\arctan (y) + C_{1} = \int x^{- 2} d x + \int - 1 d x$

Step 2.3.5

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

$\arctan (y) + C_{1} = - x^{- 1} + C_{2} + \int - 1 d x$

Step 2.3.6

Apply the constant rule.

$\arctan (y) + C_{1} = - x^{- 1} + C_{2} - x + C_{3}$

Step 2.3.7

Simplify.

$\arctan (y) + C_{1} = - \frac{1}{x} - x + C_{4}$

Step 2.3.8

Reorder terms.

$\arctan (y) + C_{1} = - x - \frac{1}{x} + C_{4}$

$\arctan (y) + C_{1} = - x - \frac{1}{x} + C_{4}$

Step 2.4

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

$\arctan (y) = - x - \frac{1}{x} + K$

$\arctan (y) = - x - \frac{1}{x} + K$

Step 3

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (- x - \frac{1}{x} + K)$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$