Calculus Examples

$\int_{- 2}^{0} \frac{1}{2} x^{4} + \frac{1}{4} x^{3} - x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{0} \frac{1}{2} x^{4} d x + \int_{- 2}^{0} \frac{1}{4} x^{3} d x + \int_{- 2}^{0} - x d x$

Step 2

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

$\frac{1}{2} \int_{- 2}^{0} x^{4} d x + \int_{- 2}^{0} \frac{1}{4} x^{3} d x + \int_{- 2}^{0} - x d x$

Step 3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \int_{- 2}^{0} \frac{1}{4} x^{3} d x + \int_{- 2}^{0} - x d x$

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \frac{1}{4} \int_{- 2}^{0} x^{3} d x + \int_{- 2}^{0} - x d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \frac{1}{4} \frac{1}{4} x^{4}]_{- 2}^{0} + \int_{- 2}^{0} - x d x$

Step 6

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

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \frac{1}{4} \frac{1}{4} x^{4}]_{- 2}^{0} - \int_{- 2}^{0} x d x$

Step 7

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

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \frac{1}{4} \frac{1}{4} x^{4}]_{- 2}^{0} - (\frac{1}{2} x^{2}]_{- 2}^{0})$

Step 8

Simplify the answer.

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Step 8.1

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

$\frac{1}{2} \frac{1}{5} x^{5}]_{- 2}^{0} + \frac{1}{4} \frac{1}{4} x^{4}]_{- 2}^{0} - (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{1}{5} x^{5}$ at $0$ and at $- 2$ .

$\frac{1}{2} ((\frac{1}{5} \cdot 0^{5}) - \frac{1}{5} {(- 2)}^{5}) + \frac{1}{4} \frac{1}{4} x^{4}]_{- 2}^{0} - (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 8.2.2

Evaluate $\frac{1}{4} x^{4}$ at $0$ and at $- 2$ .

$\frac{1}{2} (\frac{1}{5} \cdot 0^{5} - \frac{1}{5} {(- 2)}^{5}) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - (\frac{x^{2}}{2}]_{- 2}^{0})$

Step 8.2.3

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

$\frac{1}{2} (\frac{1}{5} \cdot 0^{5} - \frac{1}{5} {(- 2)}^{5}) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4

Simplify.

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Step 8.2.4.1

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

$\frac{1}{2} (\frac{1}{5} \cdot 0 - \frac{1}{5} {(- 2)}^{5}) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.2

Multiply $\frac{1}{5}$ by $0$ .

$\frac{1}{2} (0 - \frac{1}{5} {(- 2)}^{5}) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.3

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

$\frac{1}{2} (0 - \frac{1}{5} \cdot - 32) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.4

Multiply $- 32$ by $- 1$ .

$\frac{1}{2} (0 + 32 (\frac{1}{5})) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.5

Combine $32$ and $\frac{1}{5}$ .

$\frac{1}{2} (0 + \frac{32}{5}) + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.6

Add $0$ and $\frac{32}{5}$ .

$\frac{1}{2} \cdot \frac{32}{5} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.7

Multiply $\frac{1}{2}$ by $\frac{32}{5}$ .

$\frac{32}{2 \cdot 5} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.8

Multiply $2$ by $5$ .

$\frac{32}{10} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.9

Cancel the common factor of $32$ and $10$ .

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Step 8.2.4.9.1

Factor $2$ out of $32$ .

$\frac{2 (16)}{10} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.9.2

Cancel the common factors.

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Step 8.2.4.9.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 16}{2 \cdot 5} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.9.2.2

Cancel the common factor.

$\frac{2 \cdot 16}{2 \cdot 5} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.9.2.3

Rewrite the expression.

$\frac{16}{5} + \frac{1}{4} (\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.10

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

$\frac{16}{5} + \frac{1}{4} (\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.11

Multiply $\frac{1}{4}$ by $0$ .

$\frac{16}{5} + \frac{1}{4} (0 - \frac{1}{4} {(- 2)}^{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.12

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

$\frac{16}{5} + \frac{1}{4} (0 - \frac{1}{4} \cdot 16) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.13

Multiply $16$ by $- 1$ .

$\frac{16}{5} + \frac{1}{4} (0 - 16 (\frac{1}{4})) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.14

Combine $- 16$ and $\frac{1}{4}$ .

$\frac{16}{5} + \frac{1}{4} (0 + \frac{- 16}{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.15

Cancel the common factor of $- 16$ and $4$ .

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Step 8.2.4.15.1

Factor $4$ out of $- 16$ .

$\frac{16}{5} + \frac{1}{4} (0 + \frac{4 \cdot - 4}{4}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.15.2

Cancel the common factors.

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Step 8.2.4.15.2.1

Factor $4$ out of $4$ .

$\frac{16}{5} + \frac{1}{4} (0 + \frac{4 \cdot - 4}{4 (1)}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.15.2.2

Cancel the common factor.

$\frac{16}{5} + \frac{1}{4} (0 + \frac{4 \cdot - 4}{4 \cdot 1}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.15.2.3

Rewrite the expression.

$\frac{16}{5} + \frac{1}{4} (0 + \frac{- 4}{1}) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.15.2.4

Divide $- 4$ by $1$ .

$\frac{16}{5} + \frac{1}{4} (0 - 4) - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.16

Subtract $4$ from $0$ .

$\frac{16}{5} + \frac{1}{4} \cdot - 4 - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.17

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

$\frac{16}{5} + \frac{- 4}{4} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.18

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

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Step 8.2.4.18.1

Factor $4$ out of $- 4$ .

$\frac{16}{5} + \frac{4 \cdot - 1}{4} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.18.2

Cancel the common factors.

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Step 8.2.4.18.2.1

Factor $4$ out of $4$ .

$\frac{16}{5} + \frac{4 \cdot - 1}{4 (1)} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.18.2.2

Cancel the common factor.

$\frac{16}{5} + \frac{4 \cdot - 1}{4 \cdot 1} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.18.2.3

Rewrite the expression.

$\frac{16}{5} + \frac{- 1}{1} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.18.2.4

Divide $- 1$ by $1$ .

$\frac{16}{5} - 1 - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.19

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{16}{5} - 1 \cdot \frac{5}{5} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.20

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{16}{5} + \frac{- 1 \cdot 5}{5} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.21

Combine the numerators over the common denominator.

$\frac{16 - 1 \cdot 5}{5} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.22

Simplify the numerator.

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Step 8.2.4.22.1

Multiply $- 1$ by $5$ .

$\frac{16 - 5}{5} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.22.2

Subtract $5$ from $16$ .

$\frac{11}{5} - ((\frac{0^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.23

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

$\frac{11}{5} - (\frac{0}{2} - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.24

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

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Step 8.2.4.24.1

Factor $2$ out of $0$ .

$\frac{11}{5} - (\frac{2 (0)}{2} - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.24.2

Cancel the common factors.

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Step 8.2.4.24.2.1

Factor $2$ out of $2$ .

$\frac{11}{5} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.24.2.2

Cancel the common factor.

$\frac{11}{5} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.24.2.3

Rewrite the expression.

$\frac{11}{5} - (\frac{0}{1} - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.24.2.4

Divide $0$ by $1$ .

$\frac{11}{5} - (0 - \frac{{(- 2)}^{2}}{2})$

Step 8.2.4.25

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

$\frac{11}{5} - (0 - \frac{4}{2})$

Step 8.2.4.26

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

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Step 8.2.4.26.1

Factor $2$ out of $4$ .

$\frac{11}{5} - (0 - \frac{2 \cdot 2}{2})$

Step 8.2.4.26.2

Cancel the common factors.

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Step 8.2.4.26.2.1

Factor $2$ out of $2$ .

$\frac{11}{5} - (0 - \frac{2 \cdot 2}{2 (1)})$

Step 8.2.4.26.2.2

Cancel the common factor.

$\frac{11}{5} - (0 - \frac{2 \cdot 2}{2 \cdot 1})$

Step 8.2.4.26.2.3

Rewrite the expression.

$\frac{11}{5} - (0 - \frac{2}{1})$

Step 8.2.4.26.2.4

Divide $2$ by $1$ .

$\frac{11}{5} - (0 - 1 \cdot 2)$

Step 8.2.4.27

Multiply $- 1$ by $2$ .

$\frac{11}{5} - (0 - 2)$

Step 8.2.4.28

Subtract $2$ from $0$ .

$\frac{11}{5} - - 2$

Step 8.2.4.29

Multiply $- 1$ by $- 2$ .

$\frac{11}{5} + 2$

Step 8.2.4.30

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{11}{5} + 2 \cdot \frac{5}{5}$

Step 8.2.4.31

Combine $2$ and $\frac{5}{5}$ .

$\frac{11}{5} + \frac{2 \cdot 5}{5}$

Step 8.2.4.32

Combine the numerators over the common denominator.

$\frac{11 + 2 \cdot 5}{5}$

Step 8.2.4.33

Simplify the numerator.

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Step 8.2.4.33.1

Multiply $2$ by $5$ .

$\frac{11 + 10}{5}$

Step 8.2.4.33.2

Add $11$ and $10$ .

$\frac{21}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{21}{5}$

Decimal Form:

$4.2$

Mixed Number Form:

$4 \frac{1}{5}$

Step 10