Calculus Examples

$\int_{- 2}^{- 1} u - \frac{1}{u^{2}} d u$

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Evaluate $- u^{- 1}$ at $- 1$ and at $- 2$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $- 1$ to the power of $2$ .

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

Step 6.3.2

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

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

Step 6.3.3

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

$\frac{1}{2} - \frac{1}{2} \cdot 4 - (- {(- 1)}^{- 1} + {(- 2)}^{- 1})$

Step 6.3.4

Multiply $4$ by $- 1$ .

$\frac{1}{2} - 4 (\frac{1}{2}) - (- {(- 1)}^{- 1} + {(- 2)}^{- 1})$

Step 6.3.5

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{1}{2} + \frac{- 4}{2} - (- {(- 1)}^{- 1} + {(- 2)}^{- 1})$

Step 6.3.6

Cancel the common factor of $- 4$ and $2$ .

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Step 6.3.6.1

Factor $2$ out of $- 4$ .

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

Step 6.3.6.2

Cancel the common factors.

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Step 6.3.6.2.1

Factor $2$ out of $2$ .

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

Step 6.3.6.2.2

Cancel the common factor.

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

Step 6.3.6.2.3

Rewrite the expression.

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

Step 6.3.6.2.4

Divide $- 2$ by $1$ .

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

Step 6.3.7

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

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

Step 6.3.8

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

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

Step 6.3.9

Combine the numerators over the common denominator.

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

Step 6.3.10

Simplify the numerator.

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Step 6.3.10.1

Multiply $- 2$ by $2$ .

$\frac{1 - 4}{2} - (- {(- 1)}^{- 1} + {(- 2)}^{- 1})$

Step 6.3.10.2

Subtract $4$ from $1$ .

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

Step 6.3.11

Move the negative in front of the fraction.

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

Step 6.3.12

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

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

Step 6.3.13

Move the negative one from the denominator of $\frac{1}{- 1}$ .

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

Step 6.3.14

Multiply $- 1$ by $1$ .

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

Step 6.3.15

Multiply $- 1$ by $- 1$ .

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

Step 6.3.16

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

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

Step 6.3.17

Move the negative in front of the fraction.

$- \frac{3}{2} - (1 - \frac{1}{2})$

Step 6.3.18

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

$- \frac{3}{2} - (\frac{2}{2} - \frac{1}{2})$

Step 6.3.19

Combine the numerators over the common denominator.

$- \frac{3}{2} - \frac{2 - 1}{2}$

Step 6.3.20

Subtract $1$ from $2$ .

$- \frac{3}{2} - \frac{1}{2}$

Step 6.3.21

Combine the numerators over the common denominator.

$\frac{- 3 - 1}{2}$

Step 6.3.22

Subtract $1$ from $- 3$ .

$\frac{- 4}{2}$

Step 6.3.23

Cancel the common factor of $- 4$ and $2$ .

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Step 6.3.23.1

Factor $2$ out of $- 4$ .

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

Step 6.3.23.2

Cancel the common factors.

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Step 6.3.23.2.1

Factor $2$ out of $2$ .

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

Step 6.3.23.2.2

Cancel the common factor.

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

Step 6.3.23.2.3

Rewrite the expression.

$\frac{- 2}{1}$

Step 6.3.23.2.4

Divide $- 2$ by $1$ .

$- 2$

Step 7