Calculus Examples

$\int_{- 4}^{3} (5 - x^{2}) - (x - 7) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 4}^{3} 5 d x + \int_{- 4}^{3} - x^{2} d x + \int_{- 4}^{3} - (x - 7) d x$

Step 2

Apply the constant rule.

$5 x]_{- 4}^{3} + \int_{- 4}^{3} - x^{2} d x + \int_{- 4}^{3} - (x - 7) d x$

Step 3

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

$5 x]_{- 4}^{3} - \int_{- 4}^{3} x^{2} d x + \int_{- 4}^{3} - (x - 7) d x$

Step 4

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

$5 x]_{- 4}^{3} - (\frac{1}{3} x^{3}]_{- 4}^{3}) + \int_{- 4}^{3} - (x - 7) d x$

Step 5

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

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) + \int_{- 4}^{3} - (x - 7) d x$

Step 6

Multiply $- (x - 7)$ .

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) + \int_{- 4}^{3} - x - - 7 d x$

Step 7

Multiply $- 1$ by $- 7$ .

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) + \int_{- 4}^{3} - x + 7 d x$

Step 8

Split the single integral into multiple integrals.

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) + \int_{- 4}^{3} - x d x + \int_{- 4}^{3} 7 d x$

Step 9

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

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - \int_{- 4}^{3} x d x + \int_{- 4}^{3} 7 d x$

Step 10

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

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{1}{2} x^{2}]_{- 4}^{3}) + \int_{- 4}^{3} 7 d x$

Step 11

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

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{x^{2}}{2}]_{- 4}^{3}) + \int_{- 4}^{3} 7 d x$

Step 12

Apply the constant rule.

$5 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{x^{2}}{2}]_{- 4}^{3}) + 7 x]_{- 4}^{3}$

Step 13

Simplify the answer.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $5 x]_{- 4}^{3}$ and $7 x]_{- 4}^{3}$ .

$5 x + 7 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{x^{2}}{2}]_{- 4}^{3})$

Step 13.1.2

Add $5 x$ and $7 x$ .

$12 x]_{- 4}^{3} - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{x^{2}}{2}]_{- 4}^{3})$

Step 13.2

Substitute and simplify.

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Step 13.2.1

Evaluate $12 x$ at $3$ and at $- 4$ .

$(12 \cdot 3) - 12 \cdot - 4 - (\frac{x^{3}}{3}]_{- 4}^{3}) - (\frac{x^{2}}{2}]_{- 4}^{3})$

Step 13.2.2

Evaluate $\frac{x^{3}}{3}$ at $3$ and at $- 4$ .

$(12 \cdot 3) - 12 \cdot - 4 - ((\frac{3^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - (\frac{x^{2}}{2}]_{- 4}^{3})$

Step 13.2.3

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $- 4$ .

$(12 \cdot 3) - 12 \cdot - 4 - ((\frac{3^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4

Simplify.

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Step 13.2.4.1

Multiply $12$ by $3$ .

$36 - 12 \cdot - 4 - ((\frac{3^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.2

Multiply $- 12$ by $- 4$ .

$36 + 48 - ((\frac{3^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.3

Add $36$ and $48$ .

$84 - ((\frac{3^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.4

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

$84 - (\frac{27}{3} - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.5

Cancel the common factor of $27$ and $3$ .

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Step 13.2.4.5.1

Factor $3$ out of $27$ .

$84 - (\frac{3 \cdot 9}{3} - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.5.2

Cancel the common factors.

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Step 13.2.4.5.2.1

Factor $3$ out of $3$ .

$84 - (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.5.2.2

Cancel the common factor.

$84 - (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.5.2.3

Rewrite the expression.

$84 - (\frac{9}{1} - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.5.2.4

Divide $9$ by $1$ .

$84 - (9 - \frac{{(- 4)}^{3}}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.6

Raise $- 4$ to the power of $3$ .

$84 - (9 - \frac{- 64}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.7

Move the negative in front of the fraction.

$84 - (9 - - \frac{64}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.8

Multiply $- 1$ by $- 1$ .

$84 - (9 + 1 (\frac{64}{3})) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.9

Multiply $\frac{64}{3}$ by $1$ .

$84 - (9 + \frac{64}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.10

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$84 - (9 \cdot \frac{3}{3} + \frac{64}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.11

Combine $9$ and $\frac{3}{3}$ .

$84 - (\frac{9 \cdot 3}{3} + \frac{64}{3}) - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.12

Combine the numerators over the common denominator.

$84 - \frac{9 \cdot 3 + 64}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.13

Simplify the numerator.

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Step 13.2.4.13.1

Multiply $9$ by $3$ .

$84 - \frac{27 + 64}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.13.2

Add $27$ and $64$ .

$84 - \frac{91}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.14

To write $84$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$84 \cdot \frac{3}{3} - \frac{91}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.15

Combine $84$ and $\frac{3}{3}$ .

$\frac{84 \cdot 3}{3} - \frac{91}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.16

Combine the numerators over the common denominator.

$\frac{84 \cdot 3 - 91}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.17

Simplify the numerator.

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Step 13.2.4.17.1

Multiply $84$ by $3$ .

$\frac{252 - 91}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.17.2

Subtract $91$ from $252$ .

$\frac{161}{3} - ((\frac{3^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.18

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

$\frac{161}{3} - (\frac{9}{2} - \frac{{(- 4)}^{2}}{2})$

Step 13.2.4.19

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

$\frac{161}{3} - (\frac{9}{2} - \frac{16}{2})$

Step 13.2.4.20

Cancel the common factor of $16$ and $2$ .

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Step 13.2.4.20.1

Factor $2$ out of $16$ .

$\frac{161}{3} - (\frac{9}{2} - \frac{2 \cdot 8}{2})$

Step 13.2.4.20.2

Cancel the common factors.

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Step 13.2.4.20.2.1

Factor $2$ out of $2$ .

$\frac{161}{3} - (\frac{9}{2} - \frac{2 \cdot 8}{2 (1)})$

Step 13.2.4.20.2.2

Cancel the common factor.

$\frac{161}{3} - (\frac{9}{2} - \frac{2 \cdot 8}{2 \cdot 1})$

Step 13.2.4.20.2.3

Rewrite the expression.

$\frac{161}{3} - (\frac{9}{2} - \frac{8}{1})$

Step 13.2.4.20.2.4

Divide $8$ by $1$ .

$\frac{161}{3} - (\frac{9}{2} - 1 \cdot 8)$

Step 13.2.4.21

Multiply $- 1$ by $8$ .

$\frac{161}{3} - (\frac{9}{2} - 8)$

Step 13.2.4.22

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{161}{3} - (\frac{9}{2} - 8 \cdot \frac{2}{2})$

Step 13.2.4.23

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{161}{3} - (\frac{9}{2} + \frac{- 8 \cdot 2}{2})$

Step 13.2.4.24

Combine the numerators over the common denominator.

$\frac{161}{3} - \frac{9 - 8 \cdot 2}{2}$

Step 13.2.4.25

Simplify the numerator.

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Step 13.2.4.25.1

Multiply $- 8$ by $2$ .

$\frac{161}{3} - \frac{9 - 16}{2}$

Step 13.2.4.25.2

Subtract $16$ from $9$ .

$\frac{161}{3} - \frac{- 7}{2}$

Step 13.2.4.26

Move the negative in front of the fraction.

$\frac{161}{3} - - \frac{7}{2}$

Step 13.2.4.27

Multiply $- 1$ by $- 1$ .

$\frac{161}{3} + 1 (\frac{7}{2})$

Step 13.2.4.28

Multiply $\frac{7}{2}$ by $1$ .

$\frac{161}{3} + \frac{7}{2}$

Step 13.2.4.29

To write $\frac{161}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{161}{3} \cdot \frac{2}{2} + \frac{7}{2}$

Step 13.2.4.30

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{161}{3} \cdot \frac{2}{2} + \frac{7}{2} \cdot \frac{3}{3}$

Step 13.2.4.31

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.2.4.31.1

Multiply $\frac{161}{3}$ by $\frac{2}{2}$ .

$\frac{161 \cdot 2}{3 \cdot 2} + \frac{7}{2} \cdot \frac{3}{3}$

Step 13.2.4.31.2

Multiply $3$ by $2$ .

$\frac{161 \cdot 2}{6} + \frac{7}{2} \cdot \frac{3}{3}$

Step 13.2.4.31.3

Multiply $\frac{7}{2}$ by $\frac{3}{3}$ .

$\frac{161 \cdot 2}{6} + \frac{7 \cdot 3}{2 \cdot 3}$

Step 13.2.4.31.4

Multiply $2$ by $3$ .

$\frac{161 \cdot 2}{6} + \frac{7 \cdot 3}{6}$

Step 13.2.4.32

Combine the numerators over the common denominator.

$\frac{161 \cdot 2 + 7 \cdot 3}{6}$

Step 13.2.4.33

Simplify the numerator.

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Step 13.2.4.33.1

Multiply $161$ by $2$ .

$\frac{322 + 7 \cdot 3}{6}$

Step 13.2.4.33.2

Multiply $7$ by $3$ .

$\frac{322 + 21}{6}$

Step 13.2.4.33.3

Add $322$ and $21$ .

$\frac{343}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{343}{6}$

Decimal Form:

$57.1\overline{6}$

Mixed Number Form:

$57 \frac{1}{6}$

Step 15