Calculus Examples

$\int_{- 1}^{1} (3 x^{3} - x^{2} + x - 1) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

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

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

Step 11.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 1$ .

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

Step 11.2.3

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

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

Step 11.2.4

Simplify.

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Step 11.2.4.1

One to any power is one.

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

Step 11.2.4.2

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

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

Step 11.2.4.3

Combine the numerators over the common denominator.

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

Step 11.2.4.4

Subtract $1$ from $1$ .

$3 (\frac{0}{4}) - (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) - (\frac{1}{2} {(- 1)}^{2} - - 1)$

Step 11.2.4.5

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

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Step 11.2.4.5.1

Factor $4$ out of $0$ .

$3 \frac{4 (0)}{4} - (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) - (\frac{1}{2} {(- 1)}^{2} - - 1)$

Step 11.2.4.5.2

Cancel the common factors.

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Step 11.2.4.5.2.1

Factor $4$ out of $4$ .

$3 \frac{4 \cdot 0}{4 \cdot 1} - (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) - (\frac{1}{2} {(- 1)}^{2} - - 1)$

Step 11.2.4.5.2.2

Cancel the common factor.

$3 \frac{4 \cdot 0}{4 \cdot 1} - (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (\frac{1}{2} \cdot 1^{2} - 1 \cdot 1) - (\frac{1}{2} {(- 1)}^{2} - - 1)$

Step 11.2.4.5.2.3

Rewrite the expression.

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

Step 11.2.4.5.2.4

Divide $0$ by $1$ .

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

Step 11.2.4.6

Multiply $3$ by $0$ .

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

Step 11.2.4.7

One to any power is one.

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

Step 11.2.4.8

Raise $- 1$ to the power of $3$ .

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

Step 11.2.4.9

Move the negative in front of the fraction.

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

Step 11.2.4.10

Multiply $- 1$ by $- 1$ .

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

Step 11.2.4.11

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

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

Step 11.2.4.12

Combine the numerators over the common denominator.

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

Step 11.2.4.13

Add $1$ and $1$ .

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

Step 11.2.4.14

Subtract $\frac{2}{3}$ from $0$ .

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

Step 11.2.4.15

One to any power is one.

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

Step 11.2.4.16

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

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

Step 11.2.4.17

Multiply $- 1$ by $1$ .

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

Step 11.2.4.18

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

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

Step 11.2.4.19

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

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

Step 11.2.4.20

Combine the numerators over the common denominator.

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

Step 11.2.4.21

Simplify the numerator.

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Step 11.2.4.21.1

Multiply $- 1$ by $2$ .

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

Step 11.2.4.21.2

Subtract $2$ from $1$ .

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

Step 11.2.4.22

Move the negative in front of the fraction.

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

Step 11.2.4.23

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

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

Step 11.2.4.24

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

$- \frac{2}{3} - \frac{1}{2} - (\frac{1}{2} - - 1)$

Step 11.2.4.25

Multiply $- 1$ by $- 1$ .

$- \frac{2}{3} - \frac{1}{2} - (\frac{1}{2} + 1)$

Step 11.2.4.26

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

$- \frac{2}{3} - \frac{1}{2} - (\frac{1}{2} + \frac{2}{2})$

Step 11.2.4.27

Combine the numerators over the common denominator.

$- \frac{2}{3} - \frac{1}{2} - \frac{1 + 2}{2}$

Step 11.2.4.28

Add $1$ and $2$ .

$- \frac{2}{3} - \frac{1}{2} - \frac{3}{2}$

Step 11.2.4.29

Combine the numerators over the common denominator.

$- \frac{2}{3} + \frac{- 1 - 3}{2}$

Step 11.2.4.30

Subtract $3$ from $- 1$ .

$- \frac{2}{3} + \frac{- 4}{2}$

Step 11.2.4.31

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

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Step 11.2.4.31.1

Factor $2$ out of $- 4$ .

$- \frac{2}{3} + \frac{2 \cdot - 2}{2}$

Step 11.2.4.31.2

Cancel the common factors.

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Step 11.2.4.31.2.1

Factor $2$ out of $2$ .

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

Step 11.2.4.31.2.2

Cancel the common factor.

$- \frac{2}{3} + \frac{2 \cdot - 2}{2 \cdot 1}$

Step 11.2.4.31.2.3

Rewrite the expression.

$- \frac{2}{3} + \frac{- 2}{1}$

Step 11.2.4.31.2.4

Divide $- 2$ by $1$ .

$- \frac{2}{3} - 2$

Step 11.2.4.32

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 11.2.4.33

Combine $- 2$ and $\frac{3}{3}$ .

$- \frac{2}{3} + \frac{- 2 \cdot 3}{3}$

Step 11.2.4.34

Combine the numerators over the common denominator.

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

Step 11.2.4.35

Simplify the numerator.

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Step 11.2.4.35.1

Multiply $- 2$ by $3$ .

$\frac{- 2 - 6}{3}$

Step 11.2.4.35.2

Subtract $6$ from $- 2$ .

$\frac{- 8}{3}$

Step 11.2.4.36

Move the negative in front of the fraction.

$- \frac{8}{3}$

Step 12

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 13