Calculus Examples

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

Step 1

Remove parentheses.

$\int_{- 3}^{- 1} \frac{1}{x^{2}} - \frac{1}{x^{3}} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 3}^{- 1} \frac{1}{x^{2}} d x + \int_{- 3}^{- 1} - \frac{1}{x^{3}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

$\int_{- 3}^{- 1} x^{- 2} d x + \int_{- 3}^{- 1} - \frac{1}{x^{3}} d x$

Step 4

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

$- x^{- 1}]_{- 3}^{- 1} + \int_{- 3}^{- 1} - \frac{1}{x^{3}} d x$

Step 5

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

$- x^{- 1}]_{- 3}^{- 1} - \int_{- 3}^{- 1} \frac{1}{x^{3}} d x$

Step 6

Apply basic rules of exponents.

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Step 6.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- x^{- 1}]_{- 3}^{- 1} - \int_{- 3}^{- 1} {(x^{3})}^{- 1} d x$

Step 6.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 6.2.1

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

$- x^{- 1}]_{- 3}^{- 1} - \int_{- 3}^{- 1} x^{3 \cdot - 1} d x$

Step 6.2.2

Multiply $3$ by $- 1$ .

$- x^{- 1}]_{- 3}^{- 1} - \int_{- 3}^{- 1} x^{- 3} d x$

Step 7

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

$- x^{- 1}]_{- 3}^{- 1} - (- \frac{1}{2} x^{- 2}]_{- 3}^{- 1})$

Step 8

Simplify the answer.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$- x^{- 1}]_{- 3}^{- 1} - (- \frac{x^{- 2}}{2}]_{- 3}^{- 1})$

Step 8.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- x^{- 1}]_{- 3}^{- 1} - (- \frac{1}{2 x^{2}}]_{- 3}^{- 1})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $- x^{- 1}$ at $- 1$ and at $- 3$ .

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

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

Step 8.2.3.2

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

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

Step 8.2.3.3

Multiply $- 1$ by $1$ .

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

Step 8.2.3.4

Multiply $- 1$ by $- 1$ .

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

Step 8.2.3.5

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

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

Step 8.2.3.6

Move the negative in front of the fraction.

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

Step 8.2.3.7

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

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

Step 8.2.3.8

Combine the numerators over the common denominator.

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

Step 8.2.3.9

Subtract $1$ from $3$ .

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

Step 8.2.3.10

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

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

Step 8.2.3.11

Multiply $2$ by $1$ .

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

Step 8.2.3.12

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

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

Step 8.2.3.13

Multiply $2$ by $9$ .

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

Step 8.2.3.14

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 8.2.3.15

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.15.1

Multiply $\frac{1}{2}$ by $\frac{9}{9}$ .

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

Step 8.2.3.15.2

Multiply $2$ by $9$ .

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

Step 8.2.3.16

Combine the numerators over the common denominator.

$\frac{2}{3} - \frac{- 9 + 1}{18}$

Step 8.2.3.17

Add $- 9$ and $1$ .

$\frac{2}{3} - \frac{- 8}{18}$

Step 8.2.3.18

Cancel the common factor of $- 8$ and $18$ .

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Step 8.2.3.18.1

Factor $2$ out of $- 8$ .

$\frac{2}{3} - \frac{2 (- 4)}{18}$

Step 8.2.3.18.2

Cancel the common factors.

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Step 8.2.3.18.2.1

Factor $2$ out of $18$ .

$\frac{2}{3} - \frac{2 \cdot - 4}{2 \cdot 9}$

Step 8.2.3.18.2.2

Cancel the common factor.

$\frac{2}{3} - \frac{2 \cdot - 4}{2 \cdot 9}$

Step 8.2.3.18.2.3

Rewrite the expression.

$\frac{2}{3} - \frac{- 4}{9}$

Step 8.2.3.19

Move the negative in front of the fraction.

$\frac{2}{3} - - \frac{4}{9}$

Step 8.2.3.20

Multiply $- 1$ by $- 1$ .

$\frac{2}{3} + 1 (\frac{4}{9})$

Step 8.2.3.21

Multiply $\frac{4}{9}$ by $1$ .

$\frac{2}{3} + \frac{4}{9}$

Step 8.2.3.22

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

$\frac{2}{3} \cdot \frac{3}{3} + \frac{4}{9}$

Step 8.2.3.23

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.23.1

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

$\frac{2 \cdot 3}{3 \cdot 3} + \frac{4}{9}$

Step 8.2.3.23.2

Multiply $3$ by $3$ .

$\frac{2 \cdot 3}{9} + \frac{4}{9}$

Step 8.2.3.24

Combine the numerators over the common denominator.

$\frac{2 \cdot 3 + 4}{9}$

Step 8.2.3.25

Simplify the numerator.

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Step 8.2.3.25.1

Multiply $2$ by $3$ .

$\frac{6 + 4}{9}$

Step 8.2.3.25.2

Add $6$ and $4$ .

$\frac{10}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{10}{9}$

Decimal Form:

$1.\overline{1}$

Mixed Number Form:

$1 \frac{1}{9}$

Step 10