Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

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

Step 13

Substitute and simplify.

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Step 13.1

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

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

Step 13.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 2$ .

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

Step 13.3

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

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

Step 13.4

Evaluate $x$ at $1$ and at $- 2$ .

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

Step 13.5

Simplify.

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Step 13.5.1

One to any power is one.

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

Step 13.5.2

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

$4 (\frac{1}{4} - \frac{16}{4}) + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.3

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

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Step 13.5.3.1

Factor $4$ out of $16$ .

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

Step 13.5.3.2

Cancel the common factors.

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Step 13.5.3.2.1

Factor $4$ out of $4$ .

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

Step 13.5.3.2.2

Cancel the common factor.

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

Step 13.5.3.2.3

Rewrite the expression.

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

Step 13.5.3.2.4

Divide $4$ by $1$ .

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

Step 13.5.4

Multiply $- 1$ by $4$ .

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

Step 13.5.5

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

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

Step 13.5.6

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

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

Step 13.5.7

Combine the numerators over the common denominator.

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

Step 13.5.8

Simplify the numerator.

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Step 13.5.8.1

Multiply $- 4$ by $4$ .

$4 \frac{1 - 16}{4} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.8.2

Subtract $16$ from $1$ .

$4 (\frac{- 15}{4}) + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.9

Move the negative in front of the fraction.

$4 (- \frac{15}{4}) + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.10

Multiply $- 1$ by $4$ .

$- 4 (\frac{15}{4}) + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.11

Combine $- 4$ and $\frac{15}{4}$ .

$\frac{- 4 \cdot 15}{4} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.12

Multiply $- 4$ by $15$ .

$\frac{- 60}{4} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.13

Cancel the common factor of $- 60$ and $4$ .

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Step 13.5.13.1

Factor $4$ out of $- 60$ .

$\frac{4 \cdot - 15}{4} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.13.2

Cancel the common factors.

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Step 13.5.13.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 15}{4 (1)} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.13.2.2

Cancel the common factor.

$\frac{4 \cdot - 15}{4 \cdot 1} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.13.2.3

Rewrite the expression.

$\frac{- 15}{1} + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.13.2.4

Divide $- 15$ by $1$ .

$- 15 + 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.14

One to any power is one.

$- 15 + 5 (\frac{1}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.15

Raise $- 2$ to the power of $3$ .

$- 15 + 5 (\frac{1}{3} - \frac{- 8}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.16

Move the negative in front of the fraction.

$- 15 + 5 (\frac{1}{3} - - \frac{8}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.17

Multiply $- 1$ by $- 1$ .

$- 15 + 5 (\frac{1}{3} + 1 (\frac{8}{3})) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.18

Multiply $\frac{8}{3}$ by $1$ .

$- 15 + 5 (\frac{1}{3} + \frac{8}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.19

Combine the numerators over the common denominator.

$- 15 + 5 \frac{1 + 8}{3} - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.20

Add $1$ and $8$ .

$- 15 + 5 (\frac{9}{3}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.21

Cancel the common factor of $9$ and $3$ .

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Step 13.5.21.1

Factor $3$ out of $9$ .

$- 15 + 5 \frac{3 \cdot 3}{3} - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.21.2

Cancel the common factors.

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Step 13.5.21.2.1

Factor $3$ out of $3$ .

$- 15 + 5 \frac{3 \cdot 3}{3 (1)} - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.21.2.2

Cancel the common factor.

$- 15 + 5 \frac{3 \cdot 3}{3 \cdot 1} - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.21.2.3

Rewrite the expression.

$- 15 + 5 (\frac{3}{1}) - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.21.2.4

Divide $3$ by $1$ .

$- 15 + 5 \cdot 3 - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.22

Multiply $5$ by $3$ .

$- 15 + 15 - (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + 1 + 2$

Step 13.5.23

Add $- 15$ and $15$ .

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

Step 13.5.24

One to any power is one.

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

Step 13.5.25

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

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

Step 13.5.26

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

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Step 13.5.26.1

Factor $2$ out of $4$ .

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

Step 13.5.26.2

Cancel the common factors.

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Step 13.5.26.2.1

Factor $2$ out of $2$ .

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

Step 13.5.26.2.2

Cancel the common factor.

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

Step 13.5.26.2.3

Rewrite the expression.

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

Step 13.5.26.2.4

Divide $2$ by $1$ .

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

Step 13.5.27

Multiply $- 1$ by $2$ .

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

Step 13.5.28

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

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

Step 13.5.29

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

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

Step 13.5.30

Combine the numerators over the common denominator.

$0 - \frac{1 - 2 \cdot 2}{2} + 1 + 2$

Step 13.5.31

Simplify the numerator.

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Step 13.5.31.1

Multiply $- 2$ by $2$ .

$0 - \frac{1 - 4}{2} + 1 + 2$

Step 13.5.31.2

Subtract $4$ from $1$ .

$0 - \frac{- 3}{2} + 1 + 2$

Step 13.5.32

Move the negative in front of the fraction.

$0 - - \frac{3}{2} + 1 + 2$

Step 13.5.33

Multiply $- 1$ by $- 1$ .

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

Step 13.5.34

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

$0 + \frac{3}{2} + 1 + 2$

Step 13.5.35

Add $0$ and $\frac{3}{2}$ .

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

Step 13.5.36

Add $1$ and $2$ .

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

Step 13.5.37

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

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

Step 13.5.38

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

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

Step 13.5.39

Combine the numerators over the common denominator.

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

Step 13.5.40

Simplify the numerator.

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Step 13.5.40.1

Multiply $3$ by $2$ .

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

Step 13.5.40.2

Add $3$ and $6$ .

$\frac{9}{2}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$

Step 15