Calculus Examples

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

Step 1

Remove parentheses.

$\int_{- 3}^{2} \frac{1}{2} x^{2} + x + 6 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 3}^{2} \frac{1}{2} x^{2} d x + \int_{- 3}^{2} x d x + \int_{- 3}^{2} 6 d x$

Step 3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 3}^{2} x^{2} d x + \int_{- 3}^{2} x d x + \int_{- 3}^{2} 6 d x$

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

$\frac{1}{2} \frac{1}{3} x^{3}]_{- 3}^{2} + \frac{1}{2} x^{2}]_{- 3}^{2} + 6 x]_{- 3}^{2}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Combine $\frac{1}{2} x^{2}]_{- 3}^{2}$ and $6 x]_{- 3}^{2}$ .

$\frac{1}{2} \frac{1}{3} x^{3}]_{- 3}^{2} + \frac{1}{2} x^{2} + 6 x]_{- 3}^{2}$

Step 7.2

Substitute and simplify.

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Evaluate $\frac{1}{2} x^{2} + 6 x$ at $2$ and at $- 3$ .

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

Step 7.2.3

Simplify.

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

Combine $\frac{1}{3}$ and $8$ .

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

Step 7.2.3.3

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

$\frac{1}{2} (\frac{8}{3} - \frac{1}{3} \cdot - 27) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.4

Multiply $- 27$ by $- 1$ .

$\frac{1}{2} (\frac{8}{3} + 27 (\frac{1}{3})) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.5

Combine $27$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{8}{3} + \frac{27}{3}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.6

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

Tap for more steps...

Step 7.2.3.6.1

Factor $3$ out of $27$ .

$\frac{1}{2} (\frac{8}{3} + \frac{3 \cdot 9}{3}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.6.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.6.2.1

Factor $3$ out of $3$ .

$\frac{1}{2} (\frac{8}{3} + \frac{3 \cdot 9}{3 (1)}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.6.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{8}{3} + \frac{3 \cdot 9}{3 \cdot 1}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.6.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{8}{3} + \frac{9}{1}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.6.2.4

Divide $9$ by $1$ .

$\frac{1}{2} (\frac{8}{3} + 9) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.7

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

$\frac{1}{2} (\frac{8}{3} + 9 \cdot \frac{3}{3}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.8

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

$\frac{1}{2} (\frac{8}{3} + \frac{9 \cdot 3}{3}) + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.9

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{8 + 9 \cdot 3}{3} + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.10

Simplify the numerator.

Tap for more steps...

Step 7.2.3.10.1

Multiply $9$ by $3$ .

$\frac{1}{2} \cdot \frac{8 + 27}{3} + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.10.2

Add $8$ and $27$ .

$\frac{1}{2} \cdot \frac{35}{3} + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.11

Multiply $\frac{1}{2}$ by $\frac{35}{3}$ .

$\frac{35}{2 \cdot 3} + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.12

Multiply $2$ by $3$ .

$\frac{35}{6} + (\frac{1}{2} \cdot 2^{2} + 6 \cdot 2) - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.13

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

$\frac{35}{6} + \frac{1}{2} \cdot 4 + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.14

Combine $\frac{1}{2}$ and $4$ .

$\frac{35}{6} + \frac{4}{2} + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.15

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

Tap for more steps...

Step 7.2.3.15.1

Factor $2$ out of $4$ .

$\frac{35}{6} + \frac{2 \cdot 2}{2} + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.15.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.15.2.1

Factor $2$ out of $2$ .

$\frac{35}{6} + \frac{2 \cdot 2}{2 (1)} + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.15.2.2

Cancel the common factor.

$\frac{35}{6} + \frac{2 \cdot 2}{2 \cdot 1} + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.15.2.3

Rewrite the expression.

$\frac{35}{6} + \frac{2}{1} + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.15.2.4

Divide $2$ by $1$ .

$\frac{35}{6} + 2 + 6 \cdot 2 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.16

Multiply $6$ by $2$ .

$\frac{35}{6} + 2 + 12 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.17

Add $2$ and $12$ .

$\frac{35}{6} + 14 - (\frac{1}{2} {(- 3)}^{2} + 6 \cdot - 3)$

Step 7.2.3.18

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

$\frac{35}{6} + 14 - (\frac{1}{2} \cdot 9 + 6 \cdot - 3)$

Step 7.2.3.19

Combine $\frac{1}{2}$ and $9$ .

$\frac{35}{6} + 14 - (\frac{9}{2} + 6 \cdot - 3)$

Step 7.2.3.20

Multiply $6$ by $- 3$ .

$\frac{35}{6} + 14 - (\frac{9}{2} - 18)$

Step 7.2.3.21

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{35}{6} + 14 - (\frac{9}{2} - 18 \cdot \frac{2}{2})$

Step 7.2.3.22

Combine $- 18$ and $\frac{2}{2}$ .

$\frac{35}{6} + 14 - (\frac{9}{2} + \frac{- 18 \cdot 2}{2})$

Step 7.2.3.23

Combine the numerators over the common denominator.

$\frac{35}{6} + 14 - \frac{9 - 18 \cdot 2}{2}$

Step 7.2.3.24

Simplify the numerator.

Tap for more steps...

Step 7.2.3.24.1

Multiply $- 18$ by $2$ .

$\frac{35}{6} + 14 - \frac{9 - 36}{2}$

Step 7.2.3.24.2

Subtract $36$ from $9$ .

$\frac{35}{6} + 14 - \frac{- 27}{2}$

Step 7.2.3.25

Move the negative in front of the fraction.

$\frac{35}{6} + 14 - - \frac{27}{2}$

Step 7.2.3.26

Multiply $- 1$ by $- 1$ .

$\frac{35}{6} + 14 + 1 (\frac{27}{2})$

Step 7.2.3.27

Multiply $\frac{27}{2}$ by $1$ .

$\frac{35}{6} + 14 + \frac{27}{2}$

Step 7.2.3.28

To write $14$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{35}{6} + 14 \cdot \frac{2}{2} + \frac{27}{2}$

Step 7.2.3.29

Combine $14$ and $\frac{2}{2}$ .

$\frac{35}{6} + \frac{14 \cdot 2}{2} + \frac{27}{2}$

Step 7.2.3.30

Combine the numerators over the common denominator.

$\frac{35}{6} + \frac{14 \cdot 2 + 27}{2}$

Step 7.2.3.31

Simplify the numerator.

Tap for more steps...

Step 7.2.3.31.1

Multiply $14$ by $2$ .

$\frac{35}{6} + \frac{28 + 27}{2}$

Step 7.2.3.31.2

Add $28$ and $27$ .

$\frac{35}{6} + \frac{55}{2}$

Step 7.2.3.32

To write $\frac{55}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{35}{6} + \frac{55}{2} \cdot \frac{3}{3}$

Step 7.2.3.33

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

Tap for more steps...

Step 7.2.3.33.1

Multiply $\frac{55}{2}$ by $\frac{3}{3}$ .

$\frac{35}{6} + \frac{55 \cdot 3}{2 \cdot 3}$

Step 7.2.3.33.2

Multiply $2$ by $3$ .

$\frac{35}{6} + \frac{55 \cdot 3}{6}$

Step 7.2.3.34

Combine the numerators over the common denominator.

$\frac{35 + 55 \cdot 3}{6}$

Step 7.2.3.35

Simplify the numerator.

Tap for more steps...

Step 7.2.3.35.1

Multiply $55$ by $3$ .

$\frac{35 + 165}{6}$

Step 7.2.3.35.2

Add $35$ and $165$ .

$\frac{200}{6}$

Step 7.2.3.36

Cancel the common factor of $200$ and $6$ .

Tap for more steps...

Step 7.2.3.36.1

Factor $2$ out of $200$ .

$\frac{2 (100)}{6}$

Step 7.2.3.36.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.36.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 100}{2 \cdot 3}$

Step 7.2.3.36.2.2

Cancel the common factor.

$\frac{2 \cdot 100}{2 \cdot 3}$

Step 7.2.3.36.2.3

Rewrite the expression.

$\frac{100}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{100}{3}$

Decimal Form:

$33.\overline{3}$

Mixed Number Form:

$33 \frac{1}{3}$

Step 9