Calculus Examples

$\int_{1}^{3} w^{2} - 4 w + 5 d w$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} w^{2} d w + \int_{1}^{3} - 4 w d w + \int_{1}^{3} 5 d w$

Step 2

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

$\frac{1}{3} w^{3}]_{1}^{3} + \int_{1}^{3} - 4 w d w + \int_{1}^{3} 5 d w$

Step 3

Since $- 4$ is constant with respect to $w$ , move $- 4$ out of the integral.

$\frac{1}{3} w^{3}]_{1}^{3} - 4 \int_{1}^{3} w d w + \int_{1}^{3} 5 d w$

Step 4

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

$\frac{1}{3} w^{3}]_{1}^{3} - 4 (\frac{1}{2} w^{2}]_{1}^{3}) + \int_{1}^{3} 5 d w$

Step 5

Combine $\frac{1}{2}$ and $w^{2}$ .

$\frac{1}{3} w^{3}]_{1}^{3} - 4 (\frac{w^{2}}{2}]_{1}^{3}) + \int_{1}^{3} 5 d w$

Step 6

Apply the constant rule.

$\frac{1}{3} w^{3}]_{1}^{3} - 4 (\frac{w^{2}}{2}]_{1}^{3}) + 5 w]_{1}^{3}$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{3} w^{3}]_{1}^{3}$ and $5 w]_{1}^{3}$ .

$\frac{1}{3} w^{3} + 5 w]_{1}^{3} - 4 (\frac{w^{2}}{2}]_{1}^{3})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{1}{3} w^{3} + 5 w$ at $3$ and at $1$ .

$(\frac{1}{3} \cdot 3^{3} + 5 \cdot 3) - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 (\frac{w^{2}}{2}]_{1}^{3})$

Step 7.2.2

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

$(\frac{1}{3} \cdot 3^{3} + 5 \cdot 3) - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

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

$\frac{1}{3} \cdot 27 + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3

Cancel the common factor of $27$ and $3$ .

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Step 7.2.3.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.3

Rewrite the expression.

$\frac{9}{1} + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.4

Divide $9$ by $1$ .

$9 + 5 \cdot 3 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.4

Multiply $5$ by $3$ .

$9 + 15 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.5

Add $9$ and $15$ .

$24 - (\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.6

One to any power is one.

$24 - (\frac{1}{3} \cdot 1 + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.7

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

$24 - (\frac{1}{3} + 5 \cdot 1) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.8

Multiply $5$ by $1$ .

$24 - (\frac{1}{3} + 5) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.9

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$24 - (\frac{1}{3} + 5 \cdot \frac{3}{3}) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.10

Combine $5$ and $\frac{3}{3}$ .

$24 - (\frac{1}{3} + \frac{5 \cdot 3}{3}) - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.11

Combine the numerators over the common denominator.

$24 - \frac{1 + 5 \cdot 3}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.12

Simplify the numerator.

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Step 7.2.3.12.1

Multiply $5$ by $3$ .

$24 - \frac{1 + 15}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.12.2

Add $1$ and $15$ .

$24 - \frac{16}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.13

To write $24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$24 \cdot \frac{3}{3} - \frac{16}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.14

Combine $24$ and $\frac{3}{3}$ .

$\frac{24 \cdot 3}{3} - \frac{16}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{24 \cdot 3 - 16}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.16

Simplify the numerator.

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Step 7.2.3.16.1

Multiply $24$ by $3$ .

$\frac{72 - 16}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.16.2

Subtract $16$ from $72$ .

$\frac{56}{3} - 4 ((\frac{3^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.17

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

$\frac{56}{3} - 4 (\frac{9}{2} - \frac{1^{2}}{2})$

Step 7.2.3.18

One to any power is one.

$\frac{56}{3} - 4 (\frac{9}{2} - \frac{1}{2})$

Step 7.2.3.19

Combine the numerators over the common denominator.

$\frac{56}{3} - 4 \frac{9 - 1}{2}$

Step 7.2.3.20

Subtract $1$ from $9$ .

$\frac{56}{3} - 4 (\frac{8}{2})$

Step 7.2.3.21

Cancel the common factor of $8$ and $2$ .

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Step 7.2.3.21.1

Factor $2$ out of $8$ .

$\frac{56}{3} - 4 \frac{2 \cdot 4}{2}$

Step 7.2.3.21.2

Cancel the common factors.

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Step 7.2.3.21.2.1

Factor $2$ out of $2$ .

$\frac{56}{3} - 4 \frac{2 \cdot 4}{2 (1)}$

Step 7.2.3.21.2.2

Cancel the common factor.

$\frac{56}{3} - 4 \frac{2 \cdot 4}{2 \cdot 1}$

Step 7.2.3.21.2.3

Rewrite the expression.

$\frac{56}{3} - 4 (\frac{4}{1})$

Step 7.2.3.21.2.4

Divide $4$ by $1$ .

$\frac{56}{3} - 4 \cdot 4$

Step 7.2.3.22

Multiply $- 4$ by $4$ .

$\frac{56}{3} - 16$

Step 7.2.3.23

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{56}{3} - 16 \cdot \frac{3}{3}$

Step 7.2.3.24

Combine $- 16$ and $\frac{3}{3}$ .

$\frac{56}{3} + \frac{- 16 \cdot 3}{3}$

Step 7.2.3.25

Combine the numerators over the common denominator.

$\frac{56 - 16 \cdot 3}{3}$

Step 7.2.3.26

Simplify the numerator.

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Step 7.2.3.26.1

Multiply $- 16$ by $3$ .

$\frac{56 - 48}{3}$

Step 7.2.3.26.2

Subtract $48$ from $56$ .

$\frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$

Step 9