Calculus Examples

$\frac{d y}{d x} = y^{2} x^{4} - y^{2} + x^{4} - 1$

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.

$\frac{d y}{d x} = (y^{2} x^{4} - y^{2}) + x^{4} - 1$

Step 1.1.1.2

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

$\frac{d y}{d x} = y^{2} (x^{4} - 1) + 1 (x^{4} - 1)$

Step 1.1.2

Factor the polynomial by factoring out the greatest common factor, $x^{4} - 1$ .

$\frac{d y}{d x} = (x^{4} - 1) (y^{2} + 1)$

Step 1.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

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

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

Step 1.1.6.2

Factor.

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Step 1.1.6.2.1

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

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

Step 1.1.6.2.2

Remove unnecessary parentheses.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

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

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Step 1.3.1.1

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

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

Step 1.3.1.2

Cancel the common factor.

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

Step 1.3.1.3

Rewrite the expression.

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

Step 1.3.2

Expand $(x^{2} + 1) (x + 1)$ using the FOIL Method.

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Step 1.3.2.1

Apply the distributive property.

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

Step 1.3.2.2

Apply the distributive property.

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

Step 1.3.2.3

Apply the distributive property.

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

Step 1.3.3

Simplify each term.

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Step 1.3.3.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.3.1.1

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

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Step 1.3.3.1.1.1

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

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

Step 1.3.3.1.1.2

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

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

Step 1.3.3.1.2

Add $2$ and $1$ .

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Multiply $x$ by $1$ .

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

Step 1.3.3.4

Multiply $1$ by $1$ .

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

Step 1.3.4

Expand $(x^{3} + x^{2} + x + 1) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.3.5

Simplify each term.

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Step 1.3.5.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.3.5.1.1

Multiply $x^{3}$ by $x$ .

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Step 1.3.5.1.1.1

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

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

Step 1.3.5.1.1.2

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

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

Step 1.3.5.1.2

Add $3$ and $1$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} + x^{3} \cdot - 1 + x^{2} x + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.2

Move $- 1$ to the left of $x^{3}$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - 1 \cdot x^{3} + x^{2} x + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.3

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{2} x + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.5.4.1

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

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Step 1.3.5.4.1.1

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

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{2} x^{1} + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.4.1.2

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

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{2 + 1} + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.4.2

Add $2$ and $1$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} + x^{2} \cdot - 1 + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.5

Move $- 1$ to the left of $x^{2}$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - 1 \cdot x^{2} + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.6

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

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.7

Multiply $x$ by $x$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1$

Step 1.3.5.8

Move $- 1$ to the left of $x$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1$

Step 1.3.5.9

Rewrite $- 1 x$ as $- x$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x + 1 x + 1 \cdot - 1$

Step 1.3.5.10

Multiply $x$ by $1$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x + x + 1 \cdot - 1$

Step 1.3.5.11

Multiply $- 1$ by $1$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x + x - 1$

Step 1.3.6

Combine the opposite terms in $x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x + x - 1$ .

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Step 1.3.6.1

Add $- x^{3}$ and $x^{3}$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} + 0 - x^{2} + x^{2} - x + x - 1$

Step 1.3.6.2

Add $x^{4}$ and $0$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x^{2} + x^{2} - x + x - 1$

Step 1.3.6.3

Add $- x^{2}$ and $x^{2}$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} + 0 - x + x - 1$

Step 1.3.6.4

Add $x^{4}$ and $0$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - x + x - 1$

Step 1.3.6.5

Add $- x$ and $x$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} + 0 - 1$

Step 1.3.6.6

Add $x^{4}$ and $0$ .

$\frac{1}{y^{2} + 1} \frac{d y}{d x} = x^{4} - 1$

Step 1.4

Rewrite the equation.

$\frac{1}{y^{2} + 1} d y = (x^{4} - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{2} + 1} d y = \int x^{4} - 1 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Reorder $y^{2}$ and $1$ .

$\int \frac{1}{1 + y^{2}} d y = \int x^{4} - 1 d x$

Step 2.2.1.2

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

$\int \frac{1}{1^{2} + y^{2}} d y = \int x^{4} - 1 d x$

Step 2.2.2

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

$\arctan (y) + C_{1} = \int x^{4} - 1 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\arctan (y) + C_{1} = \int x^{4} d x + \int - 1 d x$

Step 2.3.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\arctan (y) + C_{1} = \frac{1}{5} x^{5} + C_{2} + \int - 1 d x$

Step 2.3.3

Apply the constant rule.

$\arctan (y) + C_{1} = \frac{1}{5} x^{5} + C_{2} - x + C_{3}$

Step 2.3.4

Simplify.

$\arctan (y) + C_{1} = \frac{1}{5} x^{5} - x + C_{4}$

Step 2.4

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

$\arctan (y) = \frac{1}{5} x^{5} - x + K$

Step 3

Solve for $y$ .

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Step 3.1

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

$y = \tan (\frac{1}{5} x^{5} - x + K)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$y = \tan (\frac{x^{5}}{5} - x + K)$