Calculus Examples

$\int_{1}^{6} t^{2} - 5 t + 6 d t$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{6} t^{2} d t + \int_{1}^{6} - 5 t d t + \int_{1}^{6} 6 d t$

Step 2

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

$\frac{1}{3} t^{3}]_{1}^{6} + \int_{1}^{6} - 5 t d t + \int_{1}^{6} 6 d t$

Step 3

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

$\frac{1}{3} t^{3}]_{1}^{6} - 5 \int_{1}^{6} t d t + \int_{1}^{6} 6 d t$

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{3} t^{3}]_{1}^{6}$ and $6 t]_{1}^{6}$ .

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{1}{3} t^{3} + 6 t$ at $6$ and at $1$ .

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

Step 7.2.2

Evaluate $\frac{t^{2}}{2}$ at $6$ and at $1$ .

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $6$ to the power of $3$ .

$\frac{1}{3} \cdot 216 + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.2

Combine $\frac{1}{3}$ and $216$ .

$\frac{216}{3} + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3

Cancel the common factor of $216$ and $3$ .

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Step 7.2.3.3.1

Factor $3$ out of $216$ .

$\frac{3 \cdot 72}{3} + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 72}{3 (1)} + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 72}{3 \cdot 1} + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.3

Rewrite the expression.

$\frac{72}{1} + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.3.2.4

Divide $72$ by $1$ .

$72 + 6 \cdot 6 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.4

Multiply $6$ by $6$ .

$72 + 36 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.5

Add $72$ and $36$ .

$108 - (\frac{1}{3} \cdot 1^{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.6

One to any power is one.

$108 - (\frac{1}{3} \cdot 1 + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.7

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

$108 - (\frac{1}{3} + 6 \cdot 1) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.8

Multiply $6$ by $1$ .

$108 - (\frac{1}{3} + 6) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.9

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$108 - (\frac{1}{3} + 6 \cdot \frac{3}{3}) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.10

Combine $6$ and $\frac{3}{3}$ .

$108 - (\frac{1}{3} + \frac{6 \cdot 3}{3}) - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.11

Combine the numerators over the common denominator.

$108 - \frac{1 + 6 \cdot 3}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.12

Simplify the numerator.

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Step 7.2.3.12.1

Multiply $6$ by $3$ .

$108 - \frac{1 + 18}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.12.2

Add $1$ and $18$ .

$108 - \frac{19}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.13

To write $108$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$108 \cdot \frac{3}{3} - \frac{19}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.14

Combine $108$ and $\frac{3}{3}$ .

$\frac{108 \cdot 3}{3} - \frac{19}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{108 \cdot 3 - 19}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.16

Simplify the numerator.

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Step 7.2.3.16.1

Multiply $108$ by $3$ .

$\frac{324 - 19}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.16.2

Subtract $19$ from $324$ .

$\frac{305}{3} - 5 ((\frac{6^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.2.3.17

Raise $6$ to the power of $2$ .

$\frac{305}{3} - 5 (\frac{36}{2} - \frac{1^{2}}{2})$

Step 7.2.3.18

Cancel the common factor of $36$ and $2$ .

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Step 7.2.3.18.1

Factor $2$ out of $36$ .

$\frac{305}{3} - 5 (\frac{2 \cdot 18}{2} - \frac{1^{2}}{2})$

Step 7.2.3.18.2

Cancel the common factors.

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Step 7.2.3.18.2.1

Factor $2$ out of $2$ .

$\frac{305}{3} - 5 (\frac{2 \cdot 18}{2 (1)} - \frac{1^{2}}{2})$

Step 7.2.3.18.2.2

Cancel the common factor.

$\frac{305}{3} - 5 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{1^{2}}{2})$

Step 7.2.3.18.2.3

Rewrite the expression.

$\frac{305}{3} - 5 (\frac{18}{1} - \frac{1^{2}}{2})$

Step 7.2.3.18.2.4

Divide $18$ by $1$ .

$\frac{305}{3} - 5 (18 - \frac{1^{2}}{2})$

Step 7.2.3.19

One to any power is one.

$\frac{305}{3} - 5 (18 - \frac{1}{2})$

Step 7.2.3.20

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

$\frac{305}{3} - 5 (18 \cdot \frac{2}{2} - \frac{1}{2})$

Step 7.2.3.21

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

$\frac{305}{3} - 5 (\frac{18 \cdot 2}{2} - \frac{1}{2})$

Step 7.2.3.22

Combine the numerators over the common denominator.

$\frac{305}{3} - 5 \frac{18 \cdot 2 - 1}{2}$

Step 7.2.3.23

Simplify the numerator.

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Step 7.2.3.23.1

Multiply $18$ by $2$ .

$\frac{305}{3} - 5 \frac{36 - 1}{2}$

Step 7.2.3.23.2

Subtract $1$ from $36$ .

$\frac{305}{3} - 5 (\frac{35}{2})$

Step 7.2.3.24

Combine $- 5$ and $\frac{35}{2}$ .

$\frac{305}{3} + \frac{- 5 \cdot 35}{2}$

Step 7.2.3.25

Multiply $- 5$ by $35$ .

$\frac{305}{3} + \frac{- 175}{2}$

Step 7.2.3.26

Move the negative in front of the fraction.

$\frac{305}{3} - \frac{175}{2}$

Step 7.2.3.27

To write $\frac{305}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{305}{3} \cdot \frac{2}{2} - \frac{175}{2}$

Step 7.2.3.28

To write $- \frac{175}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{305}{3} \cdot \frac{2}{2} - \frac{175}{2} \cdot \frac{3}{3}$

Step 7.2.3.29

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

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Step 7.2.3.29.1

Multiply $\frac{305}{3}$ by $\frac{2}{2}$ .

$\frac{305 \cdot 2}{3 \cdot 2} - \frac{175}{2} \cdot \frac{3}{3}$

Step 7.2.3.29.2

Multiply $3$ by $2$ .

$\frac{305 \cdot 2}{6} - \frac{175}{2} \cdot \frac{3}{3}$

Step 7.2.3.29.3

Multiply $\frac{175}{2}$ by $\frac{3}{3}$ .

$\frac{305 \cdot 2}{6} - \frac{175 \cdot 3}{2 \cdot 3}$

Step 7.2.3.29.4

Multiply $2$ by $3$ .

$\frac{305 \cdot 2}{6} - \frac{175 \cdot 3}{6}$

Step 7.2.3.30

Combine the numerators over the common denominator.

$\frac{305 \cdot 2 - 175 \cdot 3}{6}$

Step 7.2.3.31

Simplify the numerator.

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Step 7.2.3.31.1

Multiply $305$ by $2$ .

$\frac{610 - 175 \cdot 3}{6}$

Step 7.2.3.31.2

Multiply $- 175$ by $3$ .

$\frac{610 - 525}{6}$

Step 7.2.3.31.3

Subtract $525$ from $610$ .

$\frac{85}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{85}{6}$

Decimal Form:

$14.1\overline{6}$

Mixed Number Form:

$14 \frac{1}{6}$

Step 9