Calculus Examples

$\int_{- 1}^{0} (x^{5} - 4 x^{3} + 4 x + 4) d x$

Step 1

Remove parentheses.

$\int_{- 1}^{0} x^{5} - 4 x^{3} + 4 x + 4 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{0} x^{5} d x + \int_{- 1}^{0} - 4 x^{3} d x + \int_{- 1}^{0} 4 x d x + \int_{- 1}^{0} 4 d x$

Step 3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{1}{6} x^{6}]_{- 1}^{0} + \int_{- 1}^{0} - 4 x^{3} d x + \int_{- 1}^{0} 4 x d x + \int_{- 1}^{0} 4 d x$

Step 4

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

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 \int_{- 1}^{0} x^{3} d x + \int_{- 1}^{0} 4 x d x + \int_{- 1}^{0} 4 d x$

Step 5

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

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{1}{4} x^{4}]_{- 1}^{0}) + \int_{- 1}^{0} 4 x d x + \int_{- 1}^{0} 4 d x$

Step 6

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + \int_{- 1}^{0} 4 x d x + \int_{- 1}^{0} 4 d x$

Step 7

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

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 \int_{- 1}^{0} x d x + \int_{- 1}^{0} 4 d x$

Step 8

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

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 (\frac{1}{2} x^{2}]_{- 1}^{0}) + \int_{- 1}^{0} 4 d x$

Step 9

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

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 (\frac{x^{2}}{2}]_{- 1}^{0}) + \int_{- 1}^{0} 4 d x$

Step 10

Apply the constant rule.

$\frac{1}{6} x^{6}]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 (\frac{x^{2}}{2}]_{- 1}^{0}) + 4 x]_{- 1}^{0}$

Step 11

Simplify the answer.

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Step 11.1

Combine $\frac{1}{6} x^{6}]_{- 1}^{0}$ and $4 x]_{- 1}^{0}$ .

$\frac{1}{6} x^{6} + 4 x]_{- 1}^{0} - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{1}{6} x^{6} + 4 x$ at $0$ and at $- 1$ .

$(\frac{1}{6} \cdot 0^{6} + 4 \cdot 0) - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{x^{4}}{4}]_{- 1}^{0}) + 4 (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 11.2.2

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

$(\frac{1}{6} \cdot 0^{6} + 4 \cdot 0) - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 ((\frac{0^{4}}{4}) - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 11.2.3

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

$(\frac{1}{6} \cdot 0^{6} + 4 \cdot 0) - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4

Simplify.

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Step 11.2.4.1

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

$\frac{1}{6} \cdot 0 + 4 \cdot 0 - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.2

Multiply $\frac{1}{6}$ by $0$ .

$0 + 4 \cdot 0 - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.3

Multiply $4$ by $0$ .

$0 + 0 - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.4

Add $0$ and $0$ .

$0 - (\frac{1}{6} {(- 1)}^{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.5

Raise $- 1$ to the power of $6$ .

$0 - (\frac{1}{6} \cdot 1 + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.6

Multiply $\frac{1}{6}$ by $1$ .

$0 - (\frac{1}{6} + 4 \cdot - 1) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.7

Multiply $4$ by $- 1$ .

$0 - (\frac{1}{6} - 4) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.8

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

$0 - (\frac{1}{6} - 4 \cdot \frac{6}{6}) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.9

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

$0 - (\frac{1}{6} + \frac{- 4 \cdot 6}{6}) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.10

Combine the numerators over the common denominator.

$0 - \frac{1 - 4 \cdot 6}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.11

Simplify the numerator.

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Step 11.2.4.11.1

Multiply $- 4$ by $6$ .

$0 - \frac{1 - 24}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.11.2

Subtract $24$ from $1$ .

$0 - \frac{- 23}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.12

Move the negative in front of the fraction.

$0 - - \frac{23}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.13

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{23}{6}) - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.14

Multiply $\frac{23}{6}$ by $1$ .

$0 + \frac{23}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.15

Add $0$ and $\frac{23}{6}$ .

$\frac{23}{6} - 4 (\frac{0^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.16

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

$\frac{23}{6} - 4 (\frac{0}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.17

Cancel the common factor of $0$ and $4$ .

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Step 11.2.4.17.1

Factor $4$ out of $0$ .

$\frac{23}{6} - 4 (\frac{4 (0)}{4} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.17.2

Cancel the common factors.

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Step 11.2.4.17.2.1

Factor $4$ out of $4$ .

$\frac{23}{6} - 4 (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.17.2.2

Cancel the common factor.

$\frac{23}{6} - 4 (\frac{4 \cdot 0}{4 \cdot 1} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.17.2.3

Rewrite the expression.

$\frac{23}{6} - 4 (\frac{0}{1} - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.17.2.4

Divide $0$ by $1$ .

$\frac{23}{6} - 4 (0 - \frac{{(- 1)}^{4}}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.18

Raise $- 1$ to the power of $4$ .

$\frac{23}{6} - 4 (0 - \frac{1}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.19

Subtract $\frac{1}{4}$ from $0$ .

$\frac{23}{6} - 4 (- \frac{1}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.20

Multiply $- 1$ by $- 4$ .

$\frac{23}{6} + 4 (\frac{1}{4}) + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.21

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

$\frac{23}{6} + \frac{4}{4} + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.22

Cancel the common factor of $4$ .

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Step 11.2.4.22.1

Cancel the common factor.

$\frac{23}{6} + \frac{4}{4} + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.22.2

Rewrite the expression.

$\frac{23}{6} + 1 + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.23

Write $1$ as a fraction with a common denominator.

$\frac{23}{6} + \frac{6}{6} + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.24

Combine the numerators over the common denominator.

$\frac{23 + 6}{6} + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.25

Add $23$ and $6$ .

$\frac{29}{6} + 4 (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.26

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

$\frac{29}{6} + 4 (\frac{0}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.27

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

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Step 11.2.4.27.1

Factor $2$ out of $0$ .

$\frac{29}{6} + 4 (\frac{2 (0)}{2} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.27.2

Cancel the common factors.

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Step 11.2.4.27.2.1

Factor $2$ out of $2$ .

$\frac{29}{6} + 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.27.2.2

Cancel the common factor.

$\frac{29}{6} + 4 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.27.2.3

Rewrite the expression.

$\frac{29}{6} + 4 (\frac{0}{1} - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.27.2.4

Divide $0$ by $1$ .

$\frac{29}{6} + 4 (0 - \frac{{(- 1)}^{2}}{2})$

Step 11.2.4.28

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

$\frac{29}{6} + 4 (0 - \frac{1}{2})$

Step 11.2.4.29

Subtract $\frac{1}{2}$ from $0$ .

$\frac{29}{6} + 4 (- \frac{1}{2})$

Step 11.2.4.30

Multiply $- 1$ by $4$ .

$\frac{29}{6} - 4 (\frac{1}{2})$

Step 11.2.4.31

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

$\frac{29}{6} + \frac{- 4}{2}$

Step 11.2.4.32

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

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Step 11.2.4.32.1

Factor $2$ out of $- 4$ .

$\frac{29}{6} + \frac{2 \cdot - 2}{2}$

Step 11.2.4.32.2

Cancel the common factors.

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Step 11.2.4.32.2.1

Factor $2$ out of $2$ .

$\frac{29}{6} + \frac{2 \cdot - 2}{2 (1)}$

Step 11.2.4.32.2.2

Cancel the common factor.

$\frac{29}{6} + \frac{2 \cdot - 2}{2 \cdot 1}$

Step 11.2.4.32.2.3

Rewrite the expression.

$\frac{29}{6} + \frac{- 2}{1}$

Step 11.2.4.32.2.4

Divide $- 2$ by $1$ .

$\frac{29}{6} - 2$

Step 11.2.4.33

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{29}{6} - 2 \cdot \frac{6}{6}$

Step 11.2.4.34

Combine $- 2$ and $\frac{6}{6}$ .

$\frac{29}{6} + \frac{- 2 \cdot 6}{6}$

Step 11.2.4.35

Combine the numerators over the common denominator.

$\frac{29 - 2 \cdot 6}{6}$

Step 11.2.4.36

Simplify the numerator.

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Step 11.2.4.36.1

Multiply $- 2$ by $6$ .

$\frac{29 - 12}{6}$

Step 11.2.4.36.2

Subtract $12$ from $29$ .

$\frac{17}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{17}{6}$

Decimal Form:

$2.8\overline{3}$

Mixed Number Form:

$2 \frac{5}{6}$

Step 13