Calculus Examples

$\int_{- 6}^{- 3} u - \frac{9}{u^{2}} d u$

Step 1

Split the single integral into multiple integrals.

$\int_{- 6}^{- 3} u d u + \int_{- 6}^{- 3} - \frac{9}{u^{2}} d u$

Step 2

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

$\frac{1}{2} u^{2}]_{- 6}^{- 3} + \int_{- 6}^{- 3} - \frac{9}{u^{2}} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - \int_{- 6}^{- 3} \frac{9}{u^{2}} d u$

Step 4

Since $9$ is constant with respect to $u$ , move $9$ out of the integral.

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - (9 \int_{- 6}^{- 3} \frac{1}{u^{2}} d u)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $9$ by $- 1$ .

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - 9 \int_{- 6}^{- 3} \frac{1}{u^{2}} d u$

Step 5.2

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

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - 9 \int_{- 6}^{- 3} {(u^{2})}^{- 1} d u$

Step 5.3

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

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Step 5.3.1

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

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - 9 \int_{- 6}^{- 3} u^{2 \cdot - 1} d u$

Step 5.3.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - 9 \int_{- 6}^{- 3} u^{- 2} d u$

Step 6

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

$\frac{1}{2} u^{2}]_{- 6}^{- 3} - 9 (- u^{- 1}]_{- 6}^{- 3})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{1}{2} u^{2}$ at $- 3$ and at $- 6$ .

$(\frac{1}{2} {(- 3)}^{2}) - \frac{1}{2} {(- 6)}^{2} - 9 (- u^{- 1}]_{- 6}^{- 3})$

Step 7.2

Evaluate $- u^{- 1}$ at $- 3$ and at $- 6$ .

$\frac{1}{2} {(- 3)}^{2} - \frac{1}{2} {(- 6)}^{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3

Simplify.

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Step 7.3.1

Raise $- 3$ to the power of $2$ .

$\frac{1}{2} \cdot 9 - \frac{1}{2} {(- 6)}^{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.2

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} - \frac{1}{2} {(- 6)}^{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.3

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

$\frac{9}{2} - \frac{1}{2} \cdot 36 - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.4

Multiply $36$ by $- 1$ .

$\frac{9}{2} - 36 (\frac{1}{2}) - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.5

Combine $- 36$ and $\frac{1}{2}$ .

$\frac{9}{2} + \frac{- 36}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.6

Cancel the common factor of $- 36$ and $2$ .

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Step 7.3.6.1

Factor $2$ out of $- 36$ .

$\frac{9}{2} + \frac{2 \cdot - 18}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.6.2

Cancel the common factors.

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Step 7.3.6.2.1

Factor $2$ out of $2$ .

$\frac{9}{2} + \frac{2 \cdot - 18}{2 (1)} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.6.2.2

Cancel the common factor.

$\frac{9}{2} + \frac{2 \cdot - 18}{2 \cdot 1} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.6.2.3

Rewrite the expression.

$\frac{9}{2} + \frac{- 18}{1} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.6.2.4

Divide $- 18$ by $1$ .

$\frac{9}{2} - 18 - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.7

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9}{2} - 18 \cdot \frac{2}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.8

Combine $- 18$ and $\frac{2}{2}$ .

$\frac{9}{2} + \frac{- 18 \cdot 2}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.9

Combine the numerators over the common denominator.

$\frac{9 - 18 \cdot 2}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.10

Simplify the numerator.

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Step 7.3.10.1

Multiply $- 18$ by $2$ .

$\frac{9 - 36}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.10.2

Subtract $36$ from $9$ .

$\frac{- 27}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.11

Move the negative in front of the fraction.

$- \frac{27}{2} - 9 (- {(- 3)}^{- 1} + {(- 6)}^{- 1})$

Step 7.3.12

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{27}{2} - 9 (- \frac{1}{- 3} + {(- 6)}^{- 1})$

Step 7.3.13

Move the negative in front of the fraction.

$- \frac{27}{2} - 9 (- - \frac{1}{3} + {(- 6)}^{- 1})$

Step 7.3.14

Multiply $- 1$ by $- 1$ .

$- \frac{27}{2} - 9 (1 (\frac{1}{3}) + {(- 6)}^{- 1})$

Step 7.3.15

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

$- \frac{27}{2} - 9 (\frac{1}{3} + {(- 6)}^{- 1})$

Step 7.3.16

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{27}{2} - 9 (\frac{1}{3} + \frac{1}{- 6})$

Step 7.3.17

Move the negative in front of the fraction.

$- \frac{27}{2} - 9 (\frac{1}{3} - \frac{1}{6})$

Step 7.3.18

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{27}{2} - 9 (\frac{1}{3} \cdot \frac{2}{2} - \frac{1}{6})$

Step 7.3.19

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.19.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$- \frac{27}{2} - 9 (\frac{2}{3 \cdot 2} - \frac{1}{6})$

Step 7.3.19.2

Multiply $3$ by $2$ .

$- \frac{27}{2} - 9 (\frac{2}{6} - \frac{1}{6})$

Step 7.3.20

Combine the numerators over the common denominator.

$- \frac{27}{2} - 9 \frac{2 - 1}{6}$

Step 7.3.21

Subtract $1$ from $2$ .

$- \frac{27}{2} - 9 (\frac{1}{6})$

Step 7.3.22

Combine $- 9$ and $\frac{1}{6}$ .

$- \frac{27}{2} + \frac{- 9}{6}$

Step 7.3.23

Cancel the common factor of $- 9$ and $6$ .

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Step 7.3.23.1

Factor $3$ out of $- 9$ .

$- \frac{27}{2} + \frac{3 (- 3)}{6}$

Step 7.3.23.2

Cancel the common factors.

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Step 7.3.23.2.1

Factor $3$ out of $6$ .

$- \frac{27}{2} + \frac{3 \cdot - 3}{3 \cdot 2}$

Step 7.3.23.2.2

Cancel the common factor.

$- \frac{27}{2} + \frac{3 \cdot - 3}{3 \cdot 2}$

Step 7.3.23.2.3

Rewrite the expression.

$- \frac{27}{2} + \frac{- 3}{2}$

Step 7.3.24

Move the negative in front of the fraction.

$- \frac{27}{2} - \frac{3}{2}$

Step 7.3.25

Combine the numerators over the common denominator.

$\frac{- 27 - 3}{2}$

Step 7.3.26

Subtract $3$ from $- 27$ .

$\frac{- 30}{2}$

Step 7.3.27

Cancel the common factor of $- 30$ and $2$ .

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Step 7.3.27.1

Factor $2$ out of $- 30$ .

$\frac{2 \cdot - 15}{2}$

Step 7.3.27.2

Cancel the common factors.

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Step 7.3.27.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 15}{2 (1)}$

Step 7.3.27.2.2

Cancel the common factor.

$\frac{2 \cdot - 15}{2 \cdot 1}$

Step 7.3.27.2.3

Rewrite the expression.

$\frac{- 15}{1}$

Step 7.3.27.2.4

Divide $- 15$ by $1$ .

$- 15$

Step 8