Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Apply the constant rule.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $- (x + 3)$ .

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

Step 7

Multiply $- 1$ by $3$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

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

Step 13

Simplify the answer.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $5 x]_{- 2}^{1}$ and $- 3 x]_{- 2}^{1}$ .

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

Step 13.1.2

Subtract $3 x$ from $5 x$ .

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

Step 13.2

Substitute and simplify.

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Step 13.2.1

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

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

Step 13.2.2

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

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

Step 13.2.3

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

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

Step 13.2.4

Simplify.

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Step 13.2.4.1

Multiply $2$ by $1$ .

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

Step 13.2.4.2

Multiply $- 2$ by $- 2$ .

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

Step 13.2.4.3

Add $2$ and $4$ .

$6 - ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.4

One to any power is one.

$6 - (\frac{1}{3} - \frac{{(- 2)}^{3}}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.5

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

$6 - (\frac{1}{3} - \frac{- 8}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.6

Move the negative in front of the fraction.

$6 - (\frac{1}{3} - - \frac{8}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.7

Multiply $- 1$ by $- 1$ .

$6 - (\frac{1}{3} + 1 (\frac{8}{3})) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.8

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

$6 - (\frac{1}{3} + \frac{8}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.9

Combine the numerators over the common denominator.

$6 - \frac{1 + 8}{3} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.10

Add $1$ and $8$ .

$6 - \frac{9}{3} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.11

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

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Step 13.2.4.11.1

Factor $3$ out of $9$ .

$6 - \frac{3 \cdot 3}{3} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.11.2

Cancel the common factors.

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Step 13.2.4.11.2.1

Factor $3$ out of $3$ .

$6 - \frac{3 \cdot 3}{3 (1)} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.11.2.2

Cancel the common factor.

$6 - \frac{3 \cdot 3}{3 \cdot 1} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.11.2.3

Rewrite the expression.

$6 - \frac{3}{1} - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.11.2.4

Divide $3$ by $1$ .

$6 - 1 \cdot 3 - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.12

Multiply $- 1$ by $3$ .

$6 - 3 - ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2})$

Step 13.2.4.13

Subtract $3$ from $6$ .

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

Step 13.2.4.14

One to any power is one.

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

Step 13.2.4.15

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

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

Step 13.2.4.16

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

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Step 13.2.4.16.1

Factor $2$ out of $4$ .

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

Step 13.2.4.16.2

Cancel the common factors.

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Step 13.2.4.16.2.1

Factor $2$ out of $2$ .

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

Step 13.2.4.16.2.2

Cancel the common factor.

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

Step 13.2.4.16.2.3

Rewrite the expression.

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

Step 13.2.4.16.2.4

Divide $2$ by $1$ .

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

Step 13.2.4.17

Multiply $- 1$ by $2$ .

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

Step 13.2.4.18

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

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

Step 13.2.4.19

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

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

Step 13.2.4.20

Combine the numerators over the common denominator.

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

Step 13.2.4.21

Simplify the numerator.

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Step 13.2.4.21.1

Multiply $- 2$ by $2$ .

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

Step 13.2.4.21.2

Subtract $4$ from $1$ .

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

Step 13.2.4.22

Move the negative in front of the fraction.

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

Step 13.2.4.23

Multiply $- 1$ by $- 1$ .

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

Step 13.2.4.24

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

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

Step 13.2.4.25

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

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

Step 13.2.4.26

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

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

Step 13.2.4.27

Combine the numerators over the common denominator.

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

Step 13.2.4.28

Simplify the numerator.

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Step 13.2.4.28.1

Multiply $3$ by $2$ .

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

Step 13.2.4.28.2

Add $6$ and $3$ .

$\frac{9}{2}$

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