Calculus Examples

$\frac{1}{{(2 i)}^{3}} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1

Simplify the denominator.

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Step 1.1

Apply the product rule to $2 i$ .

$\frac{1}{2^{3} i^{3}} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.2

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

$\frac{1}{8 i^{3}} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.3

Factor out $i^{2}$ .

$\frac{1}{8 (i^{2} \cdot i)} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.4

Rewrite $i^{2}$ as $- 1$ .

$\frac{1}{8 (- 1 \cdot i)} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.5

Rewrite $- 1 i$ as $- i$ .

$\frac{1}{8 \cdot - 1 i} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.6

Combine exponents.

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Step 1.6.1

Factor out negative.

$\frac{1}{- (8 i)} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 1.6.2

Multiply $8$ by $- 1$ .

$\frac{1}{- 8 i} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 2

Multiply the numerator and denominator of $\frac{1}{- 8 i}$ by the conjugate of $- 8 i$ to make the denominator real.

$\frac{1}{- 8 i} \cdot \frac{i}{i} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{1 i}{- 8 i i} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.2

Multiply $i$ by $1$ .

$\frac{i}{- 8 i i} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Add parentheses.

$\frac{i}{- 8 (i i)} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3.2

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

$\frac{i}{- 8 (i^{1} i)} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3.3

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

$\frac{i}{- 8 (i^{1} i^{1})} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3.4

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

$\frac{i}{- 8 i^{1 + 1}} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{i}{- 8 i^{2}} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 3.3.6

Rewrite $i^{2}$ as $- 1$ .

$\frac{i}{- 8 \cdot - 1} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 4

Multiply $- 8$ by $- 1$ .

$\frac{i}{8} \cdot {(e^{i x} - e^{- i x})}^{3}$

Step 5

Use the Binomial Theorem.

$\frac{i}{8} \cdot ({(e^{i x})}^{3} + 3 {(e^{i x})}^{2} (- e^{- i x}) + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply the exponents in ${(e^{i x})}^{3}$ .

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Step 6.1.1.1

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

$\frac{i}{8} \cdot (e^{i x \cdot 3} + 3 {(e^{i x})}^{2} (- e^{- i x}) + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.1.2

Move $3$ to the left of $i x$ .

$\frac{i}{8} \cdot (e^{3 i x} + 3 {(e^{i x})}^{2} (- e^{- i x}) + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.1.3

Multiply $3$ by $- 1$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 {(e^{i x})}^{2} e^{- i x} + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.4

Multiply the exponents in ${(e^{i x})}^{2}$ .

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Step 6.1.4.1

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x \cdot 2} e^{- i x} + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.4.2

Move $2$ to the left of $i x$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{2 i x} e^{- i x} + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.5

Multiply $e^{2 i x}$ by $e^{- i x}$ by adding the exponents.

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Step 6.1.5.1

Move $e^{- i x}$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 (e^{- i x} e^{2 i x}) + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.5.2

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{- i x + 2 i x} + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.5.3

Add $- i x$ and $2 i x$ .

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

Step 6.1.5.4

Multiply $i$ by $1$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{i x} {(- e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.6

Apply the product rule to $- e^{- i x}$ .

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

Step 6.1.7

Rewrite using the commutative property of multiplication.

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

Step 6.1.8

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

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

Step 6.1.9

Multiply $3$ by $1$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{i x} {(e^{- i x})}^{2} + {(- e^{- i x})}^{3})$

Step 6.1.10

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

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Step 6.1.10.1

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{i x} e^{- i x \cdot 2} + {(- e^{- i x})}^{3})$

Step 6.1.10.2

Multiply $2$ by $- 1$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{i x} e^{- 2 i x} + {(- e^{- i x})}^{3})$

Step 6.1.11

Multiply $e^{i x}$ by $e^{- 2 i x}$ by adding the exponents.

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Step 6.1.11.1

Move $e^{- 2 i x}$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 (e^{- 2 i x} e^{i x}) + {(- e^{- i x})}^{3})$

Step 6.1.11.2

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- 2 i x + i x} + {(- e^{- i x})}^{3})$

Step 6.1.11.3

Add $- 2 i x$ and $i x$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- i x} + {(- e^{- i x})}^{3})$

Step 6.1.12

Apply the product rule to $- e^{- i x}$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- i x} + {(- 1)}^{3} {(e^{- i x})}^{3})$

Step 6.1.13

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- i x} - {(e^{- i x})}^{3})$

Step 6.1.14

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

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Step 6.1.14.1

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

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- i x} - e^{- i x \cdot 3})$

Step 6.1.14.2

Multiply $3$ by $- 1$ .

$\frac{i}{8} \cdot (e^{3 i x} - 3 e^{i x} + 3 e^{- i x} - e^{- 3 i x})$

Step 6.2

Apply the distributive property.

$\frac{i}{8} e^{3 i x} + \frac{i}{8} (- 3 e^{i x}) + \frac{i}{8} (3 e^{- i x}) + \frac{i}{8} (- e^{- 3 i x})$

Step 7

Simplify.

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Step 7.1

Combine $\frac{i}{8}$ and $e^{3 i x}$ .

$\frac{i e^{3 i x}}{8} + \frac{i}{8} (- 3 e^{i x}) + \frac{i}{8} (3 e^{- i x}) + \frac{i}{8} (- e^{- 3 i x})$

Step 7.2

Multiply $\frac{i}{8} (- 3 e^{i x})$ .

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Step 7.2.1

Combine $- 3$ and $\frac{i}{8}$ .

$\frac{i e^{3 i x}}{8} + \frac{- 3 i}{8} e^{i x} + \frac{i}{8} (3 e^{- i x}) + \frac{i}{8} (- e^{- 3 i x})$

Step 7.2.2

Combine $\frac{- 3 i}{8}$ and $e^{i x}$ .

$\frac{i e^{3 i x}}{8} + \frac{- 3 i e^{i x}}{8} + \frac{i}{8} (3 e^{- i x}) + \frac{i}{8} (- e^{- 3 i x})$

Step 7.3

Multiply $\frac{i}{8} (3 e^{- i x})$ .

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Step 7.3.1

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

$\frac{i e^{3 i x}}{8} + \frac{- 3 i e^{i x}}{8} + \frac{3 i}{8} e^{- i x} + \frac{i}{8} (- e^{- 3 i x})$

Step 7.3.2

Combine $\frac{3 i}{8}$ and $e^{- i x}$ .

$\frac{i e^{3 i x}}{8} + \frac{- 3 i e^{i x}}{8} + \frac{3 i e^{- i x}}{8} + \frac{i}{8} (- e^{- 3 i x})$

Step 7.4

Combine $\frac{i}{8}$ and $e^{- 3 i x}$ .

$\frac{i e^{3 i x}}{8} + \frac{- 3 i e^{i x}}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 8

Move the negative in front of the fraction.

$\frac{i e^{3 i x}}{8} - \frac{3 i e^{i x}}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 9

Move $i$ .

$\frac{i e^{3 x i}}{8} - \frac{3 i e^{i x}}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 10

Reorder $i$ and $e^{3 x i}$ .

$\frac{e^{3 x i} i}{8} - \frac{3 i e^{i x}}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 11

Reorder $i$ and $x$ .

$\frac{e^{3 x i} i}{8} - \frac{3 i e^{x i}}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 12

Move $i$ .

$\frac{e^{3 x i} i}{8} - \frac{3 e^{x i} i}{8} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 13

Move $i$ .

$\frac{e^{3 x i} i}{8} - \frac{3 e^{x i} i}{8} + \frac{3 i e^{- 1 x i}}{8} - \frac{i e^{- 3 i x}}{8}$

Step 14

Move $i$ .

$\frac{e^{3 x i} i}{8} - \frac{3 e^{x i} i}{8} + \frac{3 e^{- 1 x i} i}{8} - \frac{i e^{- 3 i x}}{8}$

Step 15

Move $i$ .

$\frac{e^{3 x i} i}{8} - \frac{3 e^{x i} i}{8} + \frac{3 e^{- 1 x i} i}{8} - \frac{i e^{- 3 x i}}{8}$

Step 16

Reorder $i$ and $e^{- 3 x i}$ .

$\frac{e^{3 x i} i}{8} - \frac{3 e^{x i} i}{8} + \frac{3 e^{- 1 x i} i}{8} - \frac{e^{- 3 x i} i}{8}$

Step 17

Reorder factors in $e^{3 x i} i$ .

$\frac{i e^{3 x i}}{8} - \frac{3 e^{x i} i}{8} + \frac{3 e^{- 1 x i} i}{8} - \frac{e^{- 3 x i} i}{8}$

Step 18

Reorder factors in $3 e^{x i} i$ .

$\frac{i e^{3 x i}}{8} - \frac{3 i e^{x i}}{8} + \frac{3 e^{- 1 x i} i}{8} - \frac{e^{- 3 x i} i}{8}$

Step 19

Rewrite $- 1 x$ as $- x$ .

$\frac{i e^{3 x i}}{8} - \frac{3 i e^{x i}}{8} + \frac{3 e^{- x i} i}{8} - \frac{e^{- 3 x i} i}{8}$

Step 20

Reorder factors in $3 e^{- x i} i$ .

$\frac{i e^{3 x i}}{8} - \frac{3 i e^{x i}}{8} + \frac{3 i e^{- x i}}{8} - \frac{e^{- 3 x i} i}{8}$

Step 21

Reorder factors in $e^{- 3 x i} i$ .

$\frac{i e^{3 x i}}{8} - \frac{3 i e^{x i}}{8} + \frac{3 i e^{- x i}}{8} - \frac{i e^{- 3 x i}}{8}$