Calculus Examples

$\int_{- 1}^{1} (x + 2 x^{2} - x^{3} + 5 x^{4}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 11

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

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

Step 12

Simplify the answer.

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Step 12.1

Combine $\frac{1}{5}$ and $x^{5}$ .

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

Step 12.2

Substitute and simplify.

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Step 12.2.1

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

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

Step 12.2.2

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

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

Step 12.2.3

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

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

Step 12.2.4

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

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

Step 12.2.5

Simplify.

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Step 12.2.5.1

One to any power is one.

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

Step 12.2.5.2

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

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

Step 12.2.5.3

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

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

Step 12.2.5.4

Multiply $- 1$ by $1$ .

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

Step 12.2.5.5

Combine the numerators over the common denominator.

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

Step 12.2.5.6

Subtract $1$ from $1$ .

$\frac{0}{2} + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.7

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

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Step 12.2.5.7.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2} + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.7.2

Cancel the common factors.

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Step 12.2.5.7.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 0}{2 \cdot 1} + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.7.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 \cdot 1} + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.7.2.3

Rewrite the expression.

$\frac{0}{1} + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.7.2.4

Divide $0$ by $1$ .

$0 + 2 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.8

One to any power is one.

$0 + 2 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.9

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

$0 + 2 (\frac{1}{3} - \frac{- 1}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.10

Move the negative in front of the fraction.

$0 + 2 (\frac{1}{3} - - \frac{1}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.11

Multiply $- 1$ by $- 1$ .

$0 + 2 (\frac{1}{3} + 1 (\frac{1}{3})) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.12

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

$0 + 2 (\frac{1}{3} + \frac{1}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.13

Combine the numerators over the common denominator.

$0 + 2 \frac{1 + 1}{3} - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.14

Add $1$ and $1$ .

$0 + 2 (\frac{2}{3}) - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.15

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

$0 + \frac{2 \cdot 2}{3} - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.16

Multiply $2$ by $2$ .

$0 + \frac{4}{3} - (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.17

Add $0$ and $\frac{4}{3}$ .

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

Step 12.2.5.18

One to any power is one.

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

Step 12.2.5.19

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

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

Step 12.2.5.20

Combine the numerators over the common denominator.

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

Step 12.2.5.21

Subtract $1$ from $1$ .

$\frac{4}{3} - \frac{0}{4} + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.22

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

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Step 12.2.5.22.1

Factor $4$ out of $0$ .

$\frac{4}{3} - \frac{4 (0)}{4} + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.22.2

Cancel the common factors.

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Step 12.2.5.22.2.1

Factor $4$ out of $4$ .

$\frac{4}{3} - \frac{4 \cdot 0}{4 \cdot 1} + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.22.2.2

Cancel the common factor.

$\frac{4}{3} - \frac{4 \cdot 0}{4 \cdot 1} + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.22.2.3

Rewrite the expression.

$\frac{4}{3} - \frac{0}{1} + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.22.2.4

Divide $0$ by $1$ .

$\frac{4}{3} - 0 + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.23

Multiply $- 1$ by $0$ .

$\frac{4}{3} + 0 + 5 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5})$

Step 12.2.5.24

Add $\frac{4}{3}$ and $0$ .

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

Step 12.2.5.25

One to any power is one.

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

Step 12.2.5.26

Raise $- 1$ to the power of $5$ .

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

Step 12.2.5.27

Move the negative in front of the fraction.

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

Step 12.2.5.28

Multiply $- 1$ by $- 1$ .

$\frac{4}{3} + 5 (\frac{1}{5} + 1 (\frac{1}{5}))$

Step 12.2.5.29

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

$\frac{4}{3} + 5 (\frac{1}{5} + \frac{1}{5})$

Step 12.2.5.30

Combine the numerators over the common denominator.

$\frac{4}{3} + 5 \frac{1 + 1}{5}$

Step 12.2.5.31

Add $1$ and $1$ .

$\frac{4}{3} + 5 (\frac{2}{5})$

Step 12.2.5.32

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

$\frac{4}{3} + \frac{5 \cdot 2}{5}$

Step 12.2.5.33

Multiply $5$ by $2$ .

$\frac{4}{3} + \frac{10}{5}$

Step 12.2.5.34

Cancel the common factor of $10$ and $5$ .

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Step 12.2.5.34.1

Factor $5$ out of $10$ .

$\frac{4}{3} + \frac{5 \cdot 2}{5}$

Step 12.2.5.34.2

Cancel the common factors.

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Step 12.2.5.34.2.1

Factor $5$ out of $5$ .

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

Step 12.2.5.34.2.2

Cancel the common factor.

$\frac{4}{3} + \frac{5 \cdot 2}{5 \cdot 1}$

Step 12.2.5.34.2.3

Rewrite the expression.

$\frac{4}{3} + \frac{2}{1}$

Step 12.2.5.34.2.4

Divide $2$ by $1$ .

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

Step 12.2.5.35

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

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

Step 12.2.5.36

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

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

Step 12.2.5.37

Combine the numerators over the common denominator.

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

Step 12.2.5.38

Simplify the numerator.

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Step 12.2.5.38.1

Multiply $2$ by $3$ .

$\frac{4 + 6}{3}$

Step 12.2.5.38.2

Add $4$ and $6$ .

$\frac{10}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{10}{3}$

Decimal Form:

$3.\overline{3}$

Mixed Number Form:

$3 \frac{1}{3}$

Step 14