Calculus Examples

$\int_{- 5}^{0} \frac{w^{2}}{2} + 2 w - 2 d w$

Step 1

Split the single integral into multiple integrals.

$\int_{- 5}^{0} \frac{w^{2}}{2} d w + \int_{- 5}^{0} 2 w d w + \int_{- 5}^{0} - 2 d w$

Step 2

Since $\frac{1}{2}$ is constant with respect to $w$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 5}^{0} w^{2} d w + \int_{- 5}^{0} 2 w d w + \int_{- 5}^{0} - 2 d w$

Step 3

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

$\frac{1}{2} \frac{1}{3} w^{3}]_{- 5}^{0} + \int_{- 5}^{0} 2 w d w + \int_{- 5}^{0} - 2 d w$

Step 4

Since $2$ is constant with respect to $w$ , move $2$ out of the integral.

$\frac{1}{2} \frac{1}{3} w^{3}]_{- 5}^{0} + 2 \int_{- 5}^{0} w d w + \int_{- 5}^{0} - 2 d w$

Step 5

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

$\frac{1}{2} \frac{1}{3} w^{3}]_{- 5}^{0} + 2 (\frac{1}{2} w^{2}]_{- 5}^{0}) + \int_{- 5}^{0} - 2 d w$

Step 6

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

$\frac{1}{2} \frac{1}{3} w^{3}]_{- 5}^{0} + 2 (\frac{w^{2}}{2}]_{- 5}^{0}) + \int_{- 5}^{0} - 2 d w$

Step 7

Apply the constant rule.

$\frac{1}{2} \frac{1}{3} w^{3}]_{- 5}^{0} + 2 (\frac{w^{2}}{2}]_{- 5}^{0}) + - 2 w]_{- 5}^{0}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{3} w^{3}$ at $0$ and at $- 5$ .

$\frac{1}{2} ((\frac{1}{3} \cdot 0^{3}) - \frac{1}{3} {(- 5)}^{3}) + 2 (\frac{w^{2}}{2}]_{- 5}^{0}) + - 2 w]_{- 5}^{0}$

Step 8.2

Evaluate $\frac{w^{2}}{2}$ at $0$ and at $- 5$ .

$\frac{1}{2} (\frac{1}{3} \cdot 0^{3} - \frac{1}{3} {(- 5)}^{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + - 2 w]_{- 5}^{0}$

Step 8.3

Evaluate $- 2 w$ at $0$ and at $- 5$ .

$\frac{1}{2} (\frac{1}{3} \cdot 0^{3} - \frac{1}{3} {(- 5)}^{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4

Simplify.

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Step 8.4.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (\frac{1}{3} \cdot 0 - \frac{1}{3} {(- 5)}^{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.2

Multiply $\frac{1}{3}$ by $0$ .

$\frac{1}{2} (0 - \frac{1}{3} {(- 5)}^{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.3

Raise $- 5$ to the power of $3$ .

$\frac{1}{2} (0 - \frac{1}{3} \cdot - 125) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.4

Multiply $- 125$ by $- 1$ .

$\frac{1}{2} (0 + 125 (\frac{1}{3})) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.5

Combine $125$ and $\frac{1}{3}$ .

$\frac{1}{2} (0 + \frac{125}{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.6

Add $0$ and $\frac{125}{3}$ .

$\frac{1}{2} \cdot \frac{125}{3} + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.7

Multiply $\frac{1}{2}$ by $\frac{125}{3}$ .

$\frac{125}{2 \cdot 3} + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.8

Multiply $2$ by $3$ .

$\frac{125}{6} + 2 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.9

Raising $0$ to any positive power yields $0$ .

$\frac{125}{6} + 2 (\frac{0}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.10

Cancel the common factor of $0$ and $2$ .

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Step 8.4.10.1

Factor $2$ out of $0$ .

$\frac{125}{6} + 2 (\frac{2 (0)}{2} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.10.2

Cancel the common factors.

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Step 8.4.10.2.1

Factor $2$ out of $2$ .

$\frac{125}{6} + 2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.10.2.2

Cancel the common factor.

$\frac{125}{6} + 2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.10.2.3

Rewrite the expression.

$\frac{125}{6} + 2 (\frac{0}{1} - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.10.2.4

Divide $0$ by $1$ .

$\frac{125}{6} + 2 (0 - \frac{{(- 5)}^{2}}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.11

Raise $- 5$ to the power of $2$ .

$\frac{125}{6} + 2 (0 - \frac{25}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.12

Subtract $\frac{25}{2}$ from $0$ .

$\frac{125}{6} + 2 (- \frac{25}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.13

Multiply $- 1$ by $2$ .

$\frac{125}{6} - 2 (\frac{25}{2}) + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.14

Combine $- 2$ and $\frac{25}{2}$ .

$\frac{125}{6} + \frac{- 2 \cdot 25}{2} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.15

Multiply $- 2$ by $25$ .

$\frac{125}{6} + \frac{- 50}{2} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.16

Cancel the common factor of $- 50$ and $2$ .

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Step 8.4.16.1

Factor $2$ out of $- 50$ .

$\frac{125}{6} + \frac{2 \cdot - 25}{2} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.16.2

Cancel the common factors.

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Step 8.4.16.2.1

Factor $2$ out of $2$ .

$\frac{125}{6} + \frac{2 \cdot - 25}{2 (1)} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.16.2.2

Cancel the common factor.

$\frac{125}{6} + \frac{2 \cdot - 25}{2 \cdot 1} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.16.2.3

Rewrite the expression.

$\frac{125}{6} + \frac{- 25}{1} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.16.2.4

Divide $- 25$ by $1$ .

$\frac{125}{6} - 25 + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.17

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{125}{6} - 25 \cdot \frac{6}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.18

Combine $- 25$ and $\frac{6}{6}$ .

$\frac{125}{6} + \frac{- 25 \cdot 6}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.19

Combine the numerators over the common denominator.

$\frac{125 - 25 \cdot 6}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.20

Simplify the numerator.

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Step 8.4.20.1

Multiply $- 25$ by $6$ .

$\frac{125 - 150}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.20.2

Subtract $150$ from $125$ .

$\frac{- 25}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.21

Move the negative in front of the fraction.

$- \frac{25}{6} + (- 2 \cdot 0) + 2 \cdot - 5$

Step 8.4.22

Multiply $- 2$ by $0$ .

$- \frac{25}{6} + 0 + 2 \cdot - 5$

Step 8.4.23

Multiply $2$ by $- 5$ .

$- \frac{25}{6} + 0 - 10$

Step 8.4.24

Subtract $10$ from $0$ .

$- \frac{25}{6} - 10$

Step 8.4.25

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{25}{6} - 10 \cdot \frac{6}{6}$

Step 8.4.26

Combine $- 10$ and $\frac{6}{6}$ .

$- \frac{25}{6} + \frac{- 10 \cdot 6}{6}$

Step 8.4.27

Combine the numerators over the common denominator.

$\frac{- 25 - 10 \cdot 6}{6}$

Step 8.4.28

Simplify the numerator.

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Step 8.4.28.1

Multiply $- 10$ by $6$ .

$\frac{- 25 - 60}{6}$

Step 8.4.28.2

Subtract $60$ from $- 25$ .

$\frac{- 85}{6}$

Step 8.4.29

Move the negative in front of the fraction.

$- \frac{85}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{85}{6}$

Decimal Form:

$- 14.1\overline{6}$

Mixed Number Form:

$- 14 \frac{1}{6}$

Step 10