Calculus Examples

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

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

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

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

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)

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

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

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

Step 1.1.2

Factor the polynomial by factoring out the greatest common factor,

x^{4} - 1

$x^{4} - 1$ .

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

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

Step 1.1.3

Rewrite

x^{4}

$x^{4}$ as

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

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

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

Step 1.1.4

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

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)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x^{2}

$a = x^{2}$ and

b = 1

$b = 1$ .

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

$1$ as

1^{2}

$1^{2}$ .

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

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

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)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

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

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

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

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

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

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

Step 1.2

Multiply both sides by

\frac{1}{y^{2} + 1}

$\frac{1}{y^{2} + 1}$ .

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

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

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of

y^{2} + 1

$y^{2} + 1$ .

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Step 1.3.1.1

Factor

y^{2} + 1

$y^{2} + 1$ out of

(x^{2} + 1) (x + 1) (x - 1) (y^{2} + 1)

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

\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)))

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

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

$x^{5}$ .

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