Calculus Examples

$\int_{2}^{3} - x^{2} + 5 x - 6 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{3} - x^{2} d x + \int_{2}^{3} 5 x d x + \int_{2}^{3} - 6 d x$

Step 2

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

$- \int_{2}^{3} x^{2} d x + \int_{2}^{3} 5 x d x + \int_{2}^{3} - 6 d x$

Step 3

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

$- (\frac{1}{3} x^{3}]_{2}^{3}) + \int_{2}^{3} 5 x d x + \int_{2}^{3} - 6 d x$

Step 4

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

$- (\frac{x^{3}}{3}]_{2}^{3}) + \int_{2}^{3} 5 x d x + \int_{2}^{3} - 6 d x$

Step 5

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

$- (\frac{x^{3}}{3}]_{2}^{3}) + 5 \int_{2}^{3} x d x + \int_{2}^{3} - 6 d x$

Step 6

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

$- (\frac{x^{3}}{3}]_{2}^{3}) + 5 (\frac{1}{2} x^{2}]_{2}^{3}) + \int_{2}^{3} - 6 d x$

Step 7

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

$- (\frac{x^{3}}{3}]_{2}^{3}) + 5 (\frac{x^{2}}{2}]_{2}^{3}) + \int_{2}^{3} - 6 d x$

Step 8

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{2}^{3}) + 5 (\frac{x^{2}}{2}]_{2}^{3}) + - 6 x]_{2}^{3}$

Step 9

Substitute and simplify.

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Step 9.1

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

$- ((\frac{3^{3}}{3}) - \frac{2^{3}}{3}) + 5 (\frac{x^{2}}{2}]_{2}^{3}) + - 6 x]_{2}^{3}$

Step 9.2

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

$- (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + - 6 x]_{2}^{3}$

Step 9.3

Evaluate $- 6 x$ at $3$ and at $2$ .

$- (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4

Simplify.

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Step 9.4.1

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

$- (\frac{27}{3} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.2

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

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Step 9.4.2.1

Factor $3$ out of $27$ .

$- (\frac{3 \cdot 9}{3} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $3$ out of $3$ .

$- (\frac{3 \cdot 9}{3 (1)} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.2.2.2

Cancel the common factor.

$- (\frac{3 \cdot 9}{3 \cdot 1} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.2.2.3

Rewrite the expression.

$- (\frac{9}{1} - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.2.2.4

Divide $9$ by $1$ .

$- (9 - \frac{2^{3}}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.3

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

$- (9 - \frac{8}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.4

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

$- (9 \cdot \frac{3}{3} - \frac{8}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.5

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

$- (\frac{9 \cdot 3}{3} - \frac{8}{3}) + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.6

Combine the numerators over the common denominator.

$- \frac{9 \cdot 3 - 8}{3} + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.7

Simplify the numerator.

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Step 9.4.7.1

Multiply $9$ by $3$ .

$- \frac{27 - 8}{3} + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.7.2

Subtract $8$ from $27$ .

$- \frac{19}{3} + 5 (\frac{3^{2}}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.8

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

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{2^{2}}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.9

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

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{4}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.10

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

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Step 9.4.10.1

Factor $2$ out of $4$ .

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{2 \cdot 2}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.10.2

Cancel the common factors.

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Step 9.4.10.2.1

Factor $2$ out of $2$ .

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{2 \cdot 2}{2 (1)}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.10.2.2

Cancel the common factor.

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.10.2.3

Rewrite the expression.

$- \frac{19}{3} + 5 (\frac{9}{2} - \frac{2}{1}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.10.2.4

Divide $2$ by $1$ .

$- \frac{19}{3} + 5 (\frac{9}{2} - 1 \cdot 2) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.11

Multiply $- 1$ by $2$ .

$- \frac{19}{3} + 5 (\frac{9}{2} - 2) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.12

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

$- \frac{19}{3} + 5 (\frac{9}{2} - 2 \cdot \frac{2}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.13

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

$- \frac{19}{3} + 5 (\frac{9}{2} + \frac{- 2 \cdot 2}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.14

Combine the numerators over the common denominator.

$- \frac{19}{3} + 5 \frac{9 - 2 \cdot 2}{2} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.15

Simplify the numerator.

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Step 9.4.15.1

Multiply $- 2$ by $2$ .

$- \frac{19}{3} + 5 \frac{9 - 4}{2} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.15.2

Subtract $4$ from $9$ .

$- \frac{19}{3} + 5 (\frac{5}{2}) + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.16

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

$- \frac{19}{3} + \frac{5 \cdot 5}{2} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.17

Multiply $5$ by $5$ .

$- \frac{19}{3} + \frac{25}{2} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.18

To write $- \frac{19}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{19}{3} \cdot \frac{2}{2} + \frac{25}{2} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.19

To write $\frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{19}{3} \cdot \frac{2}{2} + \frac{25}{2} \cdot \frac{3}{3} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.20

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

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Step 9.4.20.1

Multiply $\frac{19}{3}$ by $\frac{2}{2}$ .

$- \frac{19 \cdot 2}{3 \cdot 2} + \frac{25}{2} \cdot \frac{3}{3} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.20.2

Multiply $3$ by $2$ .

$- \frac{19 \cdot 2}{6} + \frac{25}{2} \cdot \frac{3}{3} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.20.3

Multiply $\frac{25}{2}$ by $\frac{3}{3}$ .

$- \frac{19 \cdot 2}{6} + \frac{25 \cdot 3}{2 \cdot 3} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.20.4

Multiply $2$ by $3$ .

$- \frac{19 \cdot 2}{6} + \frac{25 \cdot 3}{6} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.21

Combine the numerators over the common denominator.

$\frac{- 19 \cdot 2 + 25 \cdot 3}{6} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.22

Simplify the numerator.

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Step 9.4.22.1

Multiply $- 19$ by $2$ .

$\frac{- 38 + 25 \cdot 3}{6} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.22.2

Multiply $25$ by $3$ .

$\frac{- 38 + 75}{6} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.22.3

Add $- 38$ and $75$ .

$\frac{37}{6} + (- 6 \cdot 3) + 6 \cdot 2$

Step 9.4.23

Multiply $- 6$ by $3$ .

$\frac{37}{6} - 18 + 6 \cdot 2$

Step 9.4.24

Multiply $6$ by $2$ .

$\frac{37}{6} - 18 + 12$

Step 9.4.25

Add $- 18$ and $12$ .

$\frac{37}{6} - 6$

Step 9.4.26

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{37}{6} - 6 \cdot \frac{6}{6}$

Step 9.4.27

Combine $- 6$ and $\frac{6}{6}$ .

$\frac{37}{6} + \frac{- 6 \cdot 6}{6}$

Step 9.4.28

Combine the numerators over the common denominator.

$\frac{37 - 6 \cdot 6}{6}$

Step 9.4.29

Simplify the numerator.

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Step 9.4.29.1

Multiply $- 6$ by $6$ .

$\frac{37 - 36}{6}$

Step 9.4.29.2

Subtract $36$ from $37$ .

$\frac{1}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$

Step 11