Calculus Examples

$\int_{- 1}^{1} {(v^{2} - 1)}^{3} d v$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Expand ${(v^{2} - 1)}^{3}$ .

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Step 2.1

Use the Binomial Theorem.

$\int_{- 1}^{1} {(v^{2})}^{3} + 3 {(v^{2})}^{2} \cdot - 1 + 3 v^{2} {(- 1)}^{2} + {(- 1)}^{3} d v$

Step 2.2

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} {(v^{2})}^{2} + 3 {(v^{2})}^{2} \cdot - 1 + 3 v^{2} {(- 1)}^{2} + {(- 1)}^{3} d v$

Step 2.3

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} (v^{2} v^{2}) + 3 {(v^{2})}^{2} \cdot - 1 + 3 v^{2} {(- 1)}^{2} + {(- 1)}^{3} d v$

Step 2.4

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} (v^{2} v^{2}) + 3 (v^{2} v^{2}) \cdot - 1 + 3 v^{2} {(- 1)}^{2} + {(- 1)}^{3} d v$

Step 2.5

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} (v^{2} v^{2}) + 3 (v^{2} v^{2}) \cdot - 1 + 3 v^{2} (- - 1) + {(- 1)}^{3} d v$

Step 2.6

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} (v^{2} v^{2}) + 3 (v^{2} v^{2}) \cdot - 1 + 3 v^{2} (- - 1) - {(- 1)}^{2} d v$

Step 2.7

Rewrite the exponentiation as a product.

$\int_{- 1}^{1} v^{2} (v^{2} v^{2}) + 3 (v^{2} v^{2}) \cdot - 1 + 3 v^{2} (- - 1) - - - 1 d v$

Step 2.8

Move $v^{2}$ .

$\int_{- 1}^{1} v^{2} v^{2} v^{2} + 3 v^{2} \cdot - 1 v^{2} + 3 v^{2} (- - 1) - - - 1 d v$

Step 2.9

Move $v^{2}$ .

$\int_{- 1}^{1} v^{2} v^{2} v^{2} + 3 \cdot - 1 v^{2} v^{2} + 3 v^{2} (- - 1) - - - 1 d v$

Step 2.10

Move $v^{2}$ .

$\int_{- 1}^{1} v^{2} v^{2} v^{2} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - - 1 d v$

Step 2.11

Move parentheses.

$\int_{- 1}^{1} v^{2} v^{2} v^{2} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.12

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

$\int_{- 1}^{1} v^{2 + 2} v^{2} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.13

Add $2$ and $2$ .

$\int_{- 1}^{1} v^{4} v^{2} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.14

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

$\int_{- 1}^{1} v^{4 + 2} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.15

Add $4$ and $2$ .

$\int_{- 1}^{1} v^{6} + 3 \cdot - 1 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.16

Multiply $3$ by $- 1$ .

$\int_{- 1}^{1} v^{6} - 3 v^{2} v^{2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.17

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

$\int_{- 1}^{1} v^{6} - 3 v^{2 + 2} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.18

Add $2$ and $2$ .

$\int_{- 1}^{1} v^{6} - 3 v^{4} + 3 \cdot - 1 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.19

Multiply $3$ by $- 1$ .

$\int_{- 1}^{1} v^{6} - 3 v^{4} - 3 \cdot - 1 v^{2} - - 1 \cdot - 1 d v$

Step 2.20

Multiply $- 3$ by $- 1$ .

$\int_{- 1}^{1} v^{6} - 3 v^{4} + 3 v^{2} - - 1 \cdot - 1 d v$

Step 2.21

Multiply $- 1$ by $- 1$ .

$\int_{- 1}^{1} v^{6} - 3 v^{4} + 3 v^{2} + 1 \cdot - 1 d v$

Step 2.22

Multiply $- 1$ by $1$ .

$\int_{- 1}^{1} v^{6} - 3 v^{4} + 3 v^{2} - 1 d v$

Step 3

Split the single integral into multiple integrals.

$\int_{- 1}^{1} v^{6} d v + \int_{- 1}^{1} - 3 v^{4} d v + \int_{- 1}^{1} 3 v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 4

By the Power Rule, the integral of $v^{6}$ with respect to $v$ is $\frac{1}{7} v^{7}$ .

$\frac{1}{7} v^{7}]_{- 1}^{1} + \int_{- 1}^{1} - 3 v^{4} d v + \int_{- 1}^{1} 3 v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 5

Since $- 3$ is constant with respect to $v$ , move $- 3$ out of the integral.

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 \int_{- 1}^{1} v^{4} d v + \int_{- 1}^{1} 3 v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 6

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

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{1}{5} v^{5}]_{- 1}^{1}) + \int_{- 1}^{1} 3 v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 7

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

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + \int_{- 1}^{1} 3 v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 8

Since $3$ is constant with respect to $v$ , move $3$ out of the integral.

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 \int_{- 1}^{1} v^{2} d v + \int_{- 1}^{1} - 1 d v$

Step 9

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

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 (\frac{1}{3} v^{3}]_{- 1}^{1}) + \int_{- 1}^{1} - 1 d v$

Step 10

Combine $\frac{1}{3}$ and $v^{3}$ .

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 (\frac{v^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} - 1 d v$

Step 11

Apply the constant rule.

$\frac{1}{7} v^{7}]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 (\frac{v^{3}}{3}]_{- 1}^{1}) + - v]_{- 1}^{1}$

Step 12

Simplify the answer.

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Step 12.1

Combine $\frac{1}{7} v^{7}]_{- 1}^{1}$ and $- v]_{- 1}^{1}$ .

$\frac{1}{7} v^{7} - v]_{- 1}^{1} - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 (\frac{v^{3}}{3}]_{- 1}^{1})$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\frac{1}{7} v^{7} - v$ at $1$ and at $- 1$ .

$(\frac{1}{7} \cdot 1^{7} - 1 \cdot 1) - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{v^{5}}{5}]_{- 1}^{1}) + 3 (\frac{v^{3}}{3}]_{- 1}^{1})$

Step 12.2.2

Evaluate $\frac{v^{5}}{5}$ at $1$ and at $- 1$ .

$(\frac{1}{7} \cdot 1^{7} - 1 \cdot 1) - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 ((\frac{1^{5}}{5}) - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{v^{3}}{3}]_{- 1}^{1})$

Step 12.2.3

Evaluate $\frac{v^{3}}{3}$ at $1$ and at $- 1$ .

$(\frac{1}{7} \cdot 1^{7} - 1 \cdot 1) - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4

Simplify.

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Step 12.2.4.1

One to any power is one.

$\frac{1}{7} \cdot 1 - 1 \cdot 1 - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.2

Multiply $\frac{1}{7}$ by $1$ .

$\frac{1}{7} - 1 \cdot 1 - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.3

Multiply $- 1$ by $1$ .

$\frac{1}{7} - 1 - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{7} - 1 \cdot \frac{7}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.5

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{1}{7} + \frac{- 1 \cdot 7}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.6

Combine the numerators over the common denominator.

$\frac{1 - 1 \cdot 7}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.7

Simplify the numerator.

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Step 12.2.4.7.1

Multiply $- 1$ by $7$ .

$\frac{1 - 7}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.7.2

Subtract $7$ from $1$ .

$\frac{- 6}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.8

Move the negative in front of the fraction.

$- \frac{6}{7} - (\frac{1}{7} {(- 1)}^{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.9

Raise $- 1$ to the power of $7$ .

$- \frac{6}{7} - (\frac{1}{7} \cdot - 1 - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.10

Combine $\frac{1}{7}$ and $- 1$ .

$- \frac{6}{7} - (\frac{- 1}{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.11

Move the negative in front of the fraction.

$- \frac{6}{7} - (- \frac{1}{7} - - 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.12

Multiply $- 1$ by $- 1$ .

$- \frac{6}{7} - (- \frac{1}{7} + 1) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.13

Write $1$ as a fraction with a common denominator.

$- \frac{6}{7} - (- \frac{1}{7} + \frac{7}{7}) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.14

Combine the numerators over the common denominator.

$- \frac{6}{7} - \frac{- 1 + 7}{7} - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.15

Add $- 1$ and $7$ .

$- \frac{6}{7} - \frac{6}{7} - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.16

Subtract $\frac{6}{7}$ from $- \frac{6}{7}$ .

$- 2 (\frac{6}{7}) - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.17

Combine $- 2$ and $\frac{6}{7}$ .

$\frac{- 2 \cdot 6}{7} - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.18

Multiply $- 2$ by $6$ .

$\frac{- 12}{7} - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.19

Move the negative in front of the fraction.

$- \frac{12}{7} - 3 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.20

One to any power is one.

$- \frac{12}{7} - 3 (\frac{1}{5} - \frac{{(- 1)}^{5}}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.21

Raise $- 1$ to the power of $5$ .

$- \frac{12}{7} - 3 (\frac{1}{5} - \frac{- 1}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.22

Move the negative in front of the fraction.

$- \frac{12}{7} - 3 (\frac{1}{5} - - \frac{1}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.23

Multiply $- 1$ by $- 1$ .

$- \frac{12}{7} - 3 (\frac{1}{5} + 1 (\frac{1}{5})) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.24

Multiply $\frac{1}{5}$ by $1$ .

$- \frac{12}{7} - 3 (\frac{1}{5} + \frac{1}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.25

Combine the numerators over the common denominator.

$- \frac{12}{7} - 3 \frac{1 + 1}{5} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.26

Add $1$ and $1$ .

$- \frac{12}{7} - 3 (\frac{2}{5}) + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.27

Combine $- 3$ and $\frac{2}{5}$ .

$- \frac{12}{7} + \frac{- 3 \cdot 2}{5} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.28

Multiply $- 3$ by $2$ .

$- \frac{12}{7} + \frac{- 6}{5} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.29

Move the negative in front of the fraction.

$- \frac{12}{7} - \frac{6}{5} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.30

To write $- \frac{12}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{12}{7} \cdot \frac{5}{5} - \frac{6}{5} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.31

To write $- \frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{12}{7} \cdot \frac{5}{5} - \frac{6}{5} \cdot \frac{7}{7} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.32

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.4.32.1

Multiply $\frac{12}{7}$ by $\frac{5}{5}$ .

$- \frac{12 \cdot 5}{7 \cdot 5} - \frac{6}{5} \cdot \frac{7}{7} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.32.2

Multiply $7$ by $5$ .

$- \frac{12 \cdot 5}{35} - \frac{6}{5} \cdot \frac{7}{7} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.32.3

Multiply $\frac{6}{5}$ by $\frac{7}{7}$ .

$- \frac{12 \cdot 5}{35} - \frac{6 \cdot 7}{5 \cdot 7} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.32.4

Multiply $5$ by $7$ .

$- \frac{12 \cdot 5}{35} - \frac{6 \cdot 7}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.33

Combine the numerators over the common denominator.

$\frac{- 12 \cdot 5 - 6 \cdot 7}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.34

Simplify the numerator.

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Step 12.2.4.34.1

Multiply $- 12$ by $5$ .

$\frac{- 60 - 6 \cdot 7}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.34.2

Multiply $- 6$ by $7$ .

$\frac{- 60 - 42}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.34.3

Subtract $42$ from $- 60$ .

$\frac{- 102}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.35

Move the negative in front of the fraction.

$- \frac{102}{35} + 3 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.36

One to any power is one.

$- \frac{102}{35} + 3 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3})$

Step 12.2.4.37

Raise $- 1$ to the power of $3$ .

$- \frac{102}{35} + 3 (\frac{1}{3} - \frac{- 1}{3})$

Step 12.2.4.38

Move the negative in front of the fraction.

$- \frac{102}{35} + 3 (\frac{1}{3} - - \frac{1}{3})$

Step 12.2.4.39

Multiply $- 1$ by $- 1$ .

$- \frac{102}{35} + 3 (\frac{1}{3} + 1 (\frac{1}{3}))$

Step 12.2.4.40

Multiply $\frac{1}{3}$ by $1$ .

$- \frac{102}{35} + 3 (\frac{1}{3} + \frac{1}{3})$

Step 12.2.4.41

Combine the numerators over the common denominator.

$- \frac{102}{35} + 3 \frac{1 + 1}{3}$

Step 12.2.4.42

Add $1$ and $1$ .

$- \frac{102}{35} + 3 (\frac{2}{3})$

Step 12.2.4.43

Combine $3$ and $\frac{2}{3}$ .

$- \frac{102}{35} + \frac{3 \cdot 2}{3}$

Step 12.2.4.44

Multiply $3$ by $2$ .

$- \frac{102}{35} + \frac{6}{3}$

Step 12.2.4.45

Cancel the common factor of $6$ and $3$ .

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Step 12.2.4.45.1

Factor $3$ out of $6$ .

$- \frac{102}{35} + \frac{3 \cdot 2}{3}$

Step 12.2.4.45.2

Cancel the common factors.

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Step 12.2.4.45.2.1

Factor $3$ out of $3$ .

$- \frac{102}{35} + \frac{3 \cdot 2}{3 (1)}$

Step 12.2.4.45.2.2

Cancel the common factor.

$- \frac{102}{35} + \frac{3 \cdot 2}{3 \cdot 1}$

Step 12.2.4.45.2.3

Rewrite the expression.

$- \frac{102}{35} + \frac{2}{1}$

Step 12.2.4.45.2.4

Divide $2$ by $1$ .

$- \frac{102}{35} + 2$

Step 12.2.4.46

To write $2$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$- \frac{102}{35} + 2 \cdot \frac{35}{35}$

Step 12.2.4.47

Combine $2$ and $\frac{35}{35}$ .

$- \frac{102}{35} + \frac{2 \cdot 35}{35}$

Step 12.2.4.48

Combine the numerators over the common denominator.

$\frac{- 102 + 2 \cdot 35}{35}$

Step 12.2.4.49

Simplify the numerator.

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Step 12.2.4.49.1

Multiply $2$ by $35$ .

$\frac{- 102 + 70}{35}$

Step 12.2.4.49.2

Add $- 102$ and $70$ .

$\frac{- 32}{35}$

Step 12.2.4.50

Move the negative in front of the fraction.

$- \frac{32}{35}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{32}{35}$

Decimal Form:

$- 0.9\overline{142857}$

Step 14