Calculus Examples

$\int_{6 x}^{7 x} \frac{u^{2} - 1}{u^{2} + 1} d u$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{d}{d x} [\int_{6 x}^{7 x} \frac{u^{2} - 1^{2}}{u^{2} + 1} d u]$

Step 1.2

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

$\frac{d}{d x} [\int_{6 x}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 2

Split the integral into two integrals where $c$ is some value between $6 x$ and $7 x$ .

$\frac{d}{d x} [\int_{6 x}^{c} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u + \int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 3

By the Sum Rule, the derivative of $\int_{6 x}^{c} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u + \int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u$ with respect to $x$ is $\frac{d}{d x} [\int_{6 x}^{c} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u] + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$ .

$\frac{d}{d x} [\int_{6 x}^{c} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u] + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 4

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{c}^{6 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u] + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 5

Take the derivative of $- \int_{c}^{6 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [6 x] (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 6

Differentiate.

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Step 6.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 6.2

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

$6 \cdot 1 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 6.3

Multiply $6$ by $1$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + \frac{d}{d x} [\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u]$

Step 7

Take the derivative of $\int_{c}^{7 x} \frac{(u + 1) (u - 1)}{u^{2} + 1} d u$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + \frac{d}{d x} [7 x] \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8

Differentiate.

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Step 8.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + 7 \frac{d}{d x} [x] \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.2

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

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + 7 \cdot 1 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3

Simplify terms.

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Step 8.3.1

Multiply $7$ by $1$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 x)}^{2} + 1}) + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.2

Factor $6$ out of $6 x$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{{(6 (x))}^{2} + 1}) + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.3

Simplify the expression.

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Step 8.3.3.1

Apply the product rule to $6 (x)$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{6^{2} x^{2} + 1}) + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.3.2

Raise $6$ to the power of $2$ .

$6 (- \frac{(6 x + 1) (6 x - 1)}{36 x^{2} + 1}) + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.3.3

Multiply $- 1$ by $6$ .

$- 6 \frac{(6 x + 1) (6 x - 1)}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.4

Combine $- 6$ and $\frac{(6 x + 1) (6 x - 1)}{36 x^{2} + 1}$ .

$\frac{- 6 ((6 x + 1) (6 x - 1))}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.5

Move the negative in front of the fraction.

$- \frac{((6) (6 x + 1)) (6 x - 1)}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 x)}^{2} + 1}$

Step 8.3.6

Factor $7$ out of $7 x$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{{(7 (x))}^{2} + 1}$

Step 8.3.7

Simplify the expression.

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Step 8.3.7.1

Apply the product rule to $7 (x)$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{7^{2} x^{2} + 1}$

Step 8.3.7.2

Raise $7$ to the power of $2$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} + 7 \frac{(7 x + 1) (7 x - 1)}{49 x^{2} + 1}$

Step 8.3.8

Combine $7$ and $\frac{(7 x + 1) (7 x - 1)}{49 x^{2} + 1}$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} + \frac{7 ((7 x + 1) (7 x - 1))}{49 x^{2} + 1}$

Step 9

To write $- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1}$ as a fraction with a common denominator, multiply by $\frac{49 x^{2} + 1}{49 x^{2} + 1}$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} \cdot \frac{49 x^{2} + 1}{49 x^{2} + 1} + \frac{7 (7 x + 1) (7 x - 1)}{49 x^{2} + 1}$

Step 10

To write $\frac{7 (7 x + 1) (7 x - 1)}{49 x^{2} + 1}$ as a fraction with a common denominator, multiply by $\frac{36 x^{2} + 1}{36 x^{2} + 1}$ .

$- \frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1} \cdot \frac{49 x^{2} + 1}{49 x^{2} + 1} + \frac{7 (7 x + 1) (7 x - 1)}{49 x^{2} + 1} \cdot \frac{36 x^{2} + 1}{36 x^{2} + 1}$

Step 11

Write each expression with a common denominator of $(36 x^{2} + 1) (49 x^{2} + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{6 (6 x + 1) (6 x - 1)}{36 x^{2} + 1}$ by $\frac{49 x^{2} + 1}{49 x^{2} + 1}$ .

$- \frac{6 (6 x + 1) (6 x - 1) (49 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)} + \frac{7 (7 x + 1) (7 x - 1)}{49 x^{2} + 1} \cdot \frac{36 x^{2} + 1}{36 x^{2} + 1}$

Step 11.2

Multiply $\frac{7 (7 x + 1) (7 x - 1)}{49 x^{2} + 1}$ by $\frac{36 x^{2} + 1}{36 x^{2} + 1}$ .

$- \frac{6 (6 x + 1) (6 x - 1) (49 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)} + \frac{7 (7 x + 1) (7 x - 1) (36 x^{2} + 1)}{(49 x^{2} + 1) (36 x^{2} + 1)}$

Step 11.3

Reorder the factors of $(49 x^{2} + 1) (36 x^{2} + 1)$ .

$- \frac{6 (6 x + 1) (6 x - 1) (49 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)} + \frac{7 (7 x + 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 12

Combine the numerators over the common denominator.

$\frac{- 6 (6 x + 1) (6 x - 1) (49 x^{2} + 1) + 7 (7 x + 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$\frac{(- 6 (6 x) - 6 \cdot 1) (6 x - 1) (49 x^{2} + 1) + 7 (7 x + 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.2

Apply the distributive property.

$\frac{(- 6 (6 x) - 6 \cdot 1) (6 x - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Simplify each term.

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Step 13.3.1.1

Simplify each term.

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Step 13.3.1.1.1

Multiply $6$ by $- 6$ .

$\frac{(- 36 x - 6 \cdot 1) (6 x - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.1.2

Multiply $- 6$ by $1$ .

$\frac{(- 36 x - 6) (6 x - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.2

Expand $(- 36 x - 6) (6 x - 1)$ using the FOIL Method.

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Step 13.3.1.2.1

Apply the distributive property.

$\frac{(- 36 x (6 x - 1) - 6 (6 x - 1)) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.2.2

Apply the distributive property.

$\frac{(- 36 x (6 x) - 36 x \cdot - 1 - 6 (6 x - 1)) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.2.3

Apply the distributive property.

$\frac{(- 36 x (6 x) - 36 x \cdot - 1 - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3

Simplify and combine like terms.

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Step 13.3.1.3.1

Simplify each term.

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Step 13.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{(- 36 \cdot 6 x \cdot x - 36 x \cdot - 1 - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 13.3.1.3.1.2.1

Move $x$ .

$\frac{(- 36 \cdot 6 (x \cdot x) - 36 x \cdot - 1 - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{(- 36 \cdot 6 x^{2} - 36 x \cdot - 1 - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.3

Multiply $- 36$ by $6$ .

$\frac{(- 216 x^{2} - 36 x \cdot - 1 - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.4

Multiply $- 1$ by $- 36$ .

$\frac{(- 216 x^{2} + 36 x - 6 (6 x) - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.5

Multiply $6$ by $- 6$ .

$\frac{(- 216 x^{2} + 36 x - 36 x - 6 \cdot - 1) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.1.6

Multiply $- 6$ by $- 1$ .

$\frac{(- 216 x^{2} + 36 x - 36 x + 6) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.2

Subtract $36 x$ from $36 x$ .

$\frac{(- 216 x^{2} + 0 + 6) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.3.3

Add $- 216 x^{2}$ and $0$ .

$\frac{(- 216 x^{2} + 6) (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.4

Expand $(- 216 x^{2} + 6) (49 x^{2} + 1)$ using the FOIL Method.

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Step 13.3.1.4.1

Apply the distributive property.

$\frac{- 216 x^{2} (49 x^{2} + 1) + 6 (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.4.2

Apply the distributive property.

$\frac{- 216 x^{2} (49 x^{2}) - 216 x^{2} \cdot 1 + 6 (49 x^{2} + 1) + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.4.3

Apply the distributive property.

$\frac{- 216 x^{2} (49 x^{2}) - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5

Simplify and combine like terms.

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Step 13.3.1.5.1

Simplify each term.

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Step 13.3.1.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 216 \cdot 49 x^{2} x^{2} - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.2

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

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Step 13.3.1.5.1.2.1

Move $x^{2}$ .

$\frac{- 216 \cdot 49 (x^{2} x^{2}) - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.2.2

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

$\frac{- 216 \cdot 49 x^{2 + 2} - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.2.3

Add $2$ and $2$ .

$\frac{- 216 \cdot 49 x^{4} - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.3

Multiply $- 216$ by $49$ .

$\frac{- 10584 x^{4} - 216 x^{2} \cdot 1 + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.4

Multiply $- 216$ by $1$ .

$\frac{- 10584 x^{4} - 216 x^{2} + 6 (49 x^{2}) + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.5

Multiply $49$ by $6$ .

$\frac{- 10584 x^{4} - 216 x^{2} + 294 x^{2} + 6 \cdot 1 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.1.6

Multiply $6$ by $1$ .

$\frac{- 10584 x^{4} - 216 x^{2} + 294 x^{2} + 6 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.5.2

Add $- 216 x^{2}$ and $294 x^{2}$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (7 (7 x) + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.6

Simplify each term.

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Step 13.3.1.6.1

Multiply $7$ by $7$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 x + 7 \cdot 1) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.6.2

Multiply $7$ by $1$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 x + 7) (7 x - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.7

Expand $(49 x + 7) (7 x - 1)$ using the FOIL Method.

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Step 13.3.1.7.1

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 x (7 x - 1) + 7 (7 x - 1)) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.7.2

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 x (7 x) + 49 x \cdot - 1 + 7 (7 x - 1)) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.7.3

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 x (7 x) + 49 x \cdot - 1 + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8

Simplify and combine like terms.

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Step 13.3.1.8.1

Simplify each term.

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Step 13.3.1.8.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 \cdot 7 x \cdot x + 49 x \cdot - 1 + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 13.3.1.8.1.2.1

Move $x$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 \cdot 7 (x \cdot x) + 49 x \cdot - 1 + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.2.2

Multiply $x$ by $x$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (49 \cdot 7 x^{2} + 49 x \cdot - 1 + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.3

Multiply $49$ by $7$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} + 49 x \cdot - 1 + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.4

Multiply $- 1$ by $49$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} - 49 x + 7 (7 x) + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.5

Multiply $7$ by $7$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} - 49 x + 49 x + 7 \cdot - 1) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.1.6

Multiply $7$ by $- 1$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} - 49 x + 49 x - 7) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.2

Add $- 49 x$ and $49 x$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} + 0 - 7) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.8.3

Add $343 x^{2}$ and $0$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + (343 x^{2} - 7) (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.9

Expand $(343 x^{2} - 7) (36 x^{2} + 1)$ using the FOIL Method.

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Step 13.3.1.9.1

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 x^{2} (36 x^{2} + 1) - 7 (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.9.2

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 x^{2} (36 x^{2}) + 343 x^{2} \cdot 1 - 7 (36 x^{2} + 1)}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.9.3

Apply the distributive property.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 x^{2} (36 x^{2}) + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10

Simplify and combine like terms.

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Step 13.3.1.10.1

Simplify each term.

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Step 13.3.1.10.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 \cdot 36 x^{2} x^{2} + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.2

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

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Step 13.3.1.10.1.2.1

Move $x^{2}$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 \cdot 36 (x^{2} x^{2}) + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.2.2

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

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 \cdot 36 x^{2 + 2} + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.2.3

Add $2$ and $2$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 343 \cdot 36 x^{4} + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.3

Multiply $343$ by $36$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 12348 x^{4} + 343 x^{2} \cdot 1 - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.4

Multiply $343$ by $1$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 12348 x^{4} + 343 x^{2} - 7 (36 x^{2}) - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.5

Multiply $36$ by $- 7$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 12348 x^{4} + 343 x^{2} - 252 x^{2} - 7 \cdot 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.1.6

Multiply $- 7$ by $1$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 12348 x^{4} + 343 x^{2} - 252 x^{2} - 7}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.1.10.2

Subtract $252 x^{2}$ from $343 x^{2}$ .

$\frac{- 10584 x^{4} + 78 x^{2} + 6 + 12348 x^{4} + 91 x^{2} - 7}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.2

Add $- 10584 x^{4}$ and $12348 x^{4}$ .

$\frac{1764 x^{4} + 78 x^{2} + 6 + 91 x^{2} - 7}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.3

Add $78 x^{2}$ and $91 x^{2}$ .

$\frac{1764 x^{4} + 169 x^{2} + 6 - 7}{(36 x^{2} + 1) (49 x^{2} + 1)}$

Step 13.3.4

Subtract $7$ from $6$ .

$\frac{1764 x^{4} + 169 x^{2} - 1}{(36 x^{2} + 1) (49 x^{2} + 1)}$