Calculus Examples

$\int_{2}^{4} \frac{w^{4} - w}{w^{3}} d w$

Step 1

Simplify the expression.

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Step 1.1

Simplify.

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Step 1.1.1

Factor $w$ out of $w^{4} - w$ .

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Step 1.1.1.1

Factor $w$ out of $w^{4}$ .

$\int_{2}^{4} \frac{w \cdot w^{3} - w}{w^{3}} d w$

Step 1.1.1.2

Factor $w$ out of $- w$ .

$\int_{2}^{4} \frac{w \cdot w^{3} + w \cdot - 1}{w^{3}} d w$

Step 1.1.1.3

Factor $w$ out of $w \cdot w^{3} + w \cdot - 1$ .

$\int_{2}^{4} \frac{w (w^{3} - 1)}{w^{3}} d w$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $w$ out of $w^{3}$ .

$\int_{2}^{4} \frac{w (w^{3} - 1)}{w \cdot w^{2}} d w$

Step 1.1.2.2

Cancel the common factor.

$\int_{2}^{4} \frac{w (w^{3} - 1)}{w \cdot w^{2}} d w$

Step 1.1.2.3

Rewrite the expression.

$\int_{2}^{4} \frac{w^{3} - 1}{w^{2}} d w$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

$\int_{2}^{4} (w^{3} - 1) {(w^{2})}^{- 1} d w$

Step 1.2.2

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

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Step 1.2.2.1

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

$\int_{2}^{4} (w^{3} - 1) w^{2 \cdot - 1} d w$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$\int_{2}^{4} (w^{3} - 1) w^{- 2} d w$

Step 2

Multiply $(w^{3} - 1) w^{- 2}$ .

$\int_{2}^{4} w^{3} w^{- 2} - 1 w^{- 2} d w$

Step 3

Simplify.

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Step 3.1

Multiply $w^{3}$ by $w^{- 2}$ by adding the exponents.

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Step 3.1.1

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

$\int_{2}^{4} w^{3 - 2} - 1 w^{- 2} d w$

Step 3.1.2

Subtract $2$ from $3$ .

$\int_{2}^{4} w^{1} - 1 w^{- 2} d w$

Step 3.2

Simplify $w^{1}$ .

$\int_{2}^{4} w - 1 w^{- 2} d w$

Step 3.3

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

$\int_{2}^{4} w - w^{- 2} d w$

Step 4

Split the single integral into multiple integrals.

$\int_{2}^{4} w d w + \int_{2}^{4} - w^{- 2} d w$

Step 5

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

$\frac{1}{2} w^{2}]_{2}^{4} + \int_{2}^{4} - w^{- 2} d w$

Step 6

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

$\frac{1}{2} w^{2}]_{2}^{4} - \int_{2}^{4} w^{- 2} d w$

Step 7

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

$\frac{1}{2} w^{2}]_{2}^{4} - (- w^{- 1}]_{2}^{4})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{2} w^{2}$ at $4$ and at $2$ .

$(\frac{1}{2} \cdot 4^{2}) - \frac{1}{2} \cdot 2^{2} - (- w^{- 1}]_{2}^{4})$

Step 8.2

Evaluate $- w^{- 1}$ at $4$ and at $2$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Raise $4$ to the power of $2$ .

$\frac{1}{2} \cdot 16 - \frac{1}{2} \cdot 2^{2} - (- 4^{- 1} + 2^{- 1})$

Step 8.3.2

Combine $\frac{1}{2}$ and $16$ .

$\frac{16}{2} - \frac{1}{2} \cdot 2^{2} - (- 4^{- 1} + 2^{- 1})$

Step 8.3.3

Cancel the common factor of $16$ and $2$ .

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Step 8.3.3.1

Factor $2$ out of $16$ .

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

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $2$ out of $2$ .

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

Step 8.3.3.2.2

Cancel the common factor.

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

Step 8.3.3.2.3

Rewrite the expression.

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

Step 8.3.3.2.4

Divide $8$ by $1$ .

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

Step 8.3.4

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

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

Step 8.3.5

Multiply $4$ by $- 1$ .

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

Step 8.3.6

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

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

Step 8.3.7

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

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Step 8.3.7.1

Factor $2$ out of $- 4$ .

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

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Factor $2$ out of $2$ .

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

Step 8.3.7.2.2

Cancel the common factor.

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

Step 8.3.7.2.3

Rewrite the expression.

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

Step 8.3.7.2.4

Divide $- 2$ by $1$ .

$8 - 2 - (- 4^{- 1} + 2^{- 1})$

Step 8.3.8

Subtract $2$ from $8$ .

$6 - (- 4^{- 1} + 2^{- 1})$

Step 8.3.9

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

$6 - (- \frac{1}{4} + 2^{- 1})$

Step 8.3.10

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

$6 - (- \frac{1}{4} + \frac{1}{2})$

Step 8.3.11

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

$6 - (- \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2})$

Step 8.3.12

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.12.1

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

$6 - (- \frac{1}{4} + \frac{2}{2 \cdot 2})$

Step 8.3.12.2

Multiply $2$ by $2$ .

$6 - (- \frac{1}{4} + \frac{2}{4})$

Step 8.3.13

Combine the numerators over the common denominator.

$6 - \frac{- 1 + 2}{4}$

Step 8.3.14

Add $- 1$ and $2$ .

$6 - \frac{1}{4}$

Step 8.3.15

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$6 \cdot \frac{4}{4} - \frac{1}{4}$

Step 8.3.16

Combine $6$ and $\frac{4}{4}$ .

$\frac{6 \cdot 4}{4} - \frac{1}{4}$

Step 8.3.17

Combine the numerators over the common denominator.

$\frac{6 \cdot 4 - 1}{4}$

Step 8.3.18

Simplify the numerator.

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Step 8.3.18.1

Multiply $6$ by $4$ .

$\frac{24 - 1}{4}$

Step 8.3.18.2

Subtract $1$ from $24$ .

$\frac{23}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{23}{4}$

Decimal Form:

$5.75$

Mixed Number Form:

$5 \frac{3}{4}$

Step 10