Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(x + \frac{1}{x})}^{2}$ as $(x + \frac{1}{x}) (x + \frac{1}{x})$ .

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

Step 1.2

Expand $(x + \frac{1}{x}) (x + \frac{1}{x})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $x$ .

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Step 1.3.1.1.1

Cancel the common factor.

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

Step 1.3.1.1.2

Rewrite the expression.

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

Step 1.3.1.2

Cancel the common factor of $x$ .

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Step 1.3.1.2.1

Cancel the common factor.

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

Step 1.3.1.2.2

Rewrite the expression.

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

Step 1.3.1.3

Multiply $\frac{1}{x} \cdot \frac{1}{x}$ .

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Step 1.3.1.3.1

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

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

Step 1.3.1.3.2

Raise $x$ to the power of $1$ .

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

Step 1.3.1.3.3

Raise $x$ to the power of $1$ .

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

Step 1.3.1.3.4

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

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

Step 1.3.1.3.5

Add $1$ and $1$ .

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

Step 1.3.2

Add $1$ and $1$ .

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

Step 2

Simplify.

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Step 2.1

Raise $x$ to the power of $1$ .

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

Step 2.2

Raise $x$ to the power of $1$ .

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

Step 2.3

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

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

Step 2.4

Add $1$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{3} x^{3}]_{1}^{2}$ and $2 x]_{1}^{2}$ .

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

Step 8.1.2

Combine $\frac{1}{3} x^{3} + 2 x]_{1}^{2}$ and $- x^{- 1}]_{1}^{2}$ .

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Simplify.

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Step 8.2.2.1

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

$\frac{1}{3} \cdot 8 + 2 \cdot 2 - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} + 2 \cdot 2 - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.3

Multiply $2$ by $2$ .

$\frac{8}{3} + 4 - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} + 4 \cdot \frac{3}{3} - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.5

Combine $4$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{4 \cdot 3}{3} - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.6

Combine the numerators over the common denominator.

$\frac{8 + 4 \cdot 3}{3} - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.7

Simplify the numerator.

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Step 8.2.2.7.1

Multiply $4$ by $3$ .

$\frac{8 + 12}{3} - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.7.2

Add $8$ and $12$ .

$\frac{20}{3} - 2^{- 1} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.8

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

$\frac{20}{3} - \frac{1}{2} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.9

To write $\frac{20}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{20}{3} \cdot \frac{2}{2} - \frac{1}{2} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.10

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

$\frac{20}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.11

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

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Step 8.2.2.11.1

Multiply $\frac{20}{3}$ by $\frac{2}{2}$ .

$\frac{20 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.11.2

Multiply $3$ by $2$ .

$\frac{20 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.11.3

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

$\frac{20 \cdot 2}{6} - \frac{3}{2 \cdot 3} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.11.4

Multiply $2$ by $3$ .

$\frac{20 \cdot 2}{6} - \frac{3}{6} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.12

Combine the numerators over the common denominator.

$\frac{20 \cdot 2 - 3}{6} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.13

Simplify the numerator.

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Step 8.2.2.13.1

Multiply $20$ by $2$ .

$\frac{40 - 3}{6} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.13.2

Subtract $3$ from $40$ .

$\frac{37}{6} - (\frac{1}{3} \cdot 1^{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.14

One to any power is one.

$\frac{37}{6} - (\frac{1}{3} \cdot 1 + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.15

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

$\frac{37}{6} - (\frac{1}{3} + 2 \cdot 1 - 1^{- 1})$

Step 8.2.2.16

Multiply $2$ by $1$ .

$\frac{37}{6} - (\frac{1}{3} + 2 - 1^{- 1})$

Step 8.2.2.17

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

$\frac{37}{6} - (\frac{1}{3} + 2 \cdot \frac{3}{3} - 1^{- 1})$

Step 8.2.2.18

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

$\frac{37}{6} - (\frac{1}{3} + \frac{2 \cdot 3}{3} - 1^{- 1})$

Step 8.2.2.19

Combine the numerators over the common denominator.

$\frac{37}{6} - (\frac{1 + 2 \cdot 3}{3} - 1^{- 1})$

Step 8.2.2.20

Simplify the numerator.

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Step 8.2.2.20.1

Multiply $2$ by $3$ .

$\frac{37}{6} - (\frac{1 + 6}{3} - 1^{- 1})$

Step 8.2.2.20.2

Add $1$ and $6$ .

$\frac{37}{6} - (\frac{7}{3} - 1^{- 1})$

Step 8.2.2.21

One to any power is one.

$\frac{37}{6} - (\frac{7}{3} - 1 \cdot 1)$

Step 8.2.2.22

Multiply $- 1$ by $1$ .

$\frac{37}{6} - (\frac{7}{3} - 1)$

Step 8.2.2.23

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

$\frac{37}{6} - (\frac{7}{3} - 1 \cdot \frac{3}{3})$

Step 8.2.2.24

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{37}{6} - (\frac{7}{3} + \frac{- 1 \cdot 3}{3})$

Step 8.2.2.25

Combine the numerators over the common denominator.

$\frac{37}{6} - \frac{7 - 1 \cdot 3}{3}$

Step 8.2.2.26

Simplify the numerator.

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Step 8.2.2.26.1

Multiply $- 1$ by $3$ .

$\frac{37}{6} - \frac{7 - 3}{3}$

Step 8.2.2.26.2

Subtract $3$ from $7$ .

$\frac{37}{6} - \frac{4}{3}$

Step 8.2.2.27

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{37}{6} - \frac{4}{3} \cdot \frac{2}{2}$

Step 8.2.2.28

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

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Step 8.2.2.28.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{37}{6} - \frac{4 \cdot 2}{3 \cdot 2}$

Step 8.2.2.28.2

Multiply $3$ by $2$ .

$\frac{37}{6} - \frac{4 \cdot 2}{6}$

Step 8.2.2.29

Combine the numerators over the common denominator.

$\frac{37 - 4 \cdot 2}{6}$

Step 8.2.2.30

Simplify the numerator.

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Step 8.2.2.30.1

Multiply $- 4$ by $2$ .

$\frac{37 - 8}{6}$

Step 8.2.2.30.2

Subtract $8$ from $37$ .

$\frac{29}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{29}{6}$

Decimal Form:

$4.8\overline{3}$

Mixed Number Form:

$4 \frac{5}{6}$

Step 10