Calculus Examples

$\frac{1}{(3 - 0) \int_{0}^{3} 0.1729 t + 0.1522 t^{2} - 0.0374 t^{3} d t}$

Step 1

Simplify.

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Step 1.1

Multiply $- 1$ by $0$ .

$\frac{1}{(3 + 0) \int_{0}^{3} 0.1729 t + 0.1522 t^{2} - 0.0374 t^{3} d t}$

Step 1.2

Add $3$ and $0$ .

$\frac{1}{3 \int_{0}^{3} 0.1729 t + 0.1522 t^{2} - 0.0374 t^{3} d t}$

Step 2

Split the single integral into multiple integrals.

$\frac{1}{3 (\int_{0}^{3} 0.1729 t d t + \int_{0}^{3} 0.1522 t^{2} d t + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 3

Since $0.1729$ is constant with respect to $t$ , move $0.1729$ out of the integral.

$\frac{1}{3 (0.1729 \int_{0}^{3} t d t + \int_{0}^{3} 0.1522 t^{2} d t + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 4

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

$\frac{1}{3 (0.1729 (\frac{1}{2} t^{2}]_{0}^{3}) + \int_{0}^{3} 0.1522 t^{2} d t + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 5

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

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + \int_{0}^{3} 0.1522 t^{2} d t + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 6

Since $0.1522$ is constant with respect to $t$ , move $0.1522$ out of the integral.

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 \int_{0}^{3} t^{2} d t + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 7

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

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 (\frac{1}{3} t^{3}]_{0}^{3}) + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 8

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

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 (\frac{t^{3}}{3}]_{0}^{3}) + \int_{0}^{3} - 0.0374 t^{3} d t)}$

Step 9

Since $- 0.0374$ is constant with respect to $t$ , move $- 0.0374$ out of the integral.

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 (\frac{t^{3}}{3}]_{0}^{3}) - 0.0374 \int_{0}^{3} t^{3} d t)}$

Step 10

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 (\frac{t^{3}}{3}]_{0}^{3}) - 0.0374 (\frac{1}{4} t^{4}]_{0}^{3}))}$

Step 11

Simplify the answer.

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Step 11.1

Combine $\frac{1}{4}$ and $t^{4}$ .

$\frac{1}{3 (0.1729 (\frac{t^{2}}{2}]_{0}^{3}) + 0.1522 (\frac{t^{3}}{3}]_{0}^{3}) - 0.0374 (\frac{t^{4}}{4}]_{0}^{3}))}$

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{t^{2}}{2}$ at $3$ and at $0$ .

$\frac{1}{3 (0.1729 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2}) + 0.1522 (\frac{t^{3}}{3}]_{0}^{3}) - 0.0374 (\frac{t^{4}}{4}]_{0}^{3}))}$

Step 11.2.2

Evaluate $\frac{t^{3}}{3}$ at $3$ and at $0$ .

$\frac{1}{3 (0.1729 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 (\frac{t^{4}}{4}]_{0}^{3}))}$

Step 11.2.3

Evaluate $\frac{t^{4}}{4}$ at $3$ and at $0$ .

$\frac{1}{3 (0.1729 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4

Simplify.

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Step 11.2.4.1

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

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{0^{2}}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.2

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{0}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.3

Cancel the common factor of $0$ and $2$ .

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Step 11.2.4.3.1

Factor $2$ out of $0$ .

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{2 (0)}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.3.2

Cancel the common factors.

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Step 11.2.4.3.2.1

Factor $2$ out of $2$ .

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.3.2.2

Cancel the common factor.

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.3.2.3

Rewrite the expression.

$\frac{1}{3 (0.1729 (\frac{9}{2} - \frac{0}{1}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.3.2.4

Divide $0$ by $1$ .

$\frac{1}{3 (0.1729 (\frac{9}{2} - 0) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.4

Multiply $- 1$ by $0$ .

$\frac{1}{3 (0.1729 (\frac{9}{2} + 0) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.5

Add $\frac{9}{2}$ and $0$ .

$\frac{1}{3 (0.1729 (\frac{9}{2}) + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.6

Combine $0.1729$ and $\frac{9}{2}$ .

$\frac{1}{3 (\frac{0.1729 \cdot 9}{2} + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.7

Multiply $0.1729$ by $9$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.8

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

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{27}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.9

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

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Step 11.2.4.9.1

Factor $3$ out of $27$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.9.2

Cancel the common factors.

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Step 11.2.4.9.2.1

Factor $3$ out of $3$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.9.2.2

Cancel the common factor.

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.9.2.3

Rewrite the expression.

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (\frac{9}{1} - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.9.2.4

Divide $9$ by $1$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{0^{3}}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.10

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{0}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.11

Cancel the common factor of $0$ and $3$ .

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Step 11.2.4.11.1

Factor $3$ out of $0$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{3 (0)}{3}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.11.2

Cancel the common factors.

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Step 11.2.4.11.2.1

Factor $3$ out of $3$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.11.2.2

Cancel the common factor.

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.11.2.3

Rewrite the expression.

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - \frac{0}{1}) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.11.2.4

Divide $0$ by $1$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 - 0) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.12

Multiply $- 1$ by $0$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 (9 + 0) - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.13

Add $9$ and $0$ .

$\frac{1}{3 (\frac{1.5561}{2} + 0.1522 \cdot 9 - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.14

Multiply $0.1522$ by $9$ .

$\frac{1}{3 (\frac{1.5561}{2} + 1.3698 - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.15

To write $1.3698$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3 (\frac{1.5561}{2} + 1.3698 \cdot \frac{2}{2} - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.16

Combine $1.3698$ and $\frac{2}{2}$ .

$\frac{1}{3 (\frac{1.5561}{2} + \frac{1.3698 \cdot 2}{2} - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.17

Combine the numerators over the common denominator.

$\frac{1}{3 (\frac{1.5561 + 1.3698 \cdot 2}{2} - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.18

Multiply $1.3698$ by $2$ .

$\frac{1}{3 (\frac{1.5561 + 2.7396}{2} - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.19

Add $1.5561$ and $2.7396$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 ((\frac{3^{4}}{4}) - \frac{0^{4}}{4}))}$

Step 11.2.4.20

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

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{0^{4}}{4}))}$

Step 11.2.4.21

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{0}{4}))}$

Step 11.2.4.22

Cancel the common factor of $0$ and $4$ .

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Step 11.2.4.22.1

Factor $4$ out of $0$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{4 (0)}{4}))}$

Step 11.2.4.22.2

Cancel the common factors.

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Step 11.2.4.22.2.1

Factor $4$ out of $4$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}))}$

Step 11.2.4.22.2.2

Cancel the common factor.

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}))}$

Step 11.2.4.22.2.3

Rewrite the expression.

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - \frac{0}{1}))}$

Step 11.2.4.22.2.4

Divide $0$ by $1$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} - 0))}$

Step 11.2.4.23

Multiply $- 1$ by $0$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4} + 0))}$

Step 11.2.4.24

Add $\frac{81}{4}$ and $0$ .

$\frac{1}{3 (\frac{4.2957}{2} - 0.0374 (\frac{81}{4}))}$

Step 11.2.4.25

Combine $- 0.0374$ and $\frac{81}{4}$ .

$\frac{1}{3 (\frac{4.2957}{2} + \frac{- 0.0374 \cdot 81}{4})}$

Step 11.2.4.26

Multiply $- 0.0374$ by $81$ .

$\frac{1}{3 (\frac{4.2957}{2} + \frac{- 3.0294}{4})}$

Step 11.2.4.27

Move the negative in front of the fraction.

$\frac{1}{3 (\frac{4.2957}{2} - \frac{3.0294}{4})}$

Step 11.2.4.28

To write $\frac{4.2957}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{3 (\frac{4.2957}{2} \cdot \frac{2}{2} - \frac{3.0294}{4})}$

Step 11.2.4.29

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.4.29.1

Multiply $\frac{4.2957}{2}$ by $\frac{2}{2}$ .

$\frac{1}{3 (\frac{4.2957 \cdot 2}{2 \cdot 2} - \frac{3.0294}{4})}$

Step 11.2.4.29.2

Multiply $2$ by $2$ .

$\frac{1}{3 (\frac{4.2957 \cdot 2}{4} - \frac{3.0294}{4})}$

Step 11.2.4.30

Combine the numerators over the common denominator.

$\frac{1}{3 \frac{4.2957 \cdot 2 - 3.0294}{4}}$

Step 11.2.4.31

Multiply $4.2957$ by $2$ .

$\frac{1}{3 \frac{8.5914 - 3.0294}{4}}$

Step 11.2.4.32

Subtract $3.0294$ from $8.5914$ .

$\frac{1}{3 (\frac{5.562}{4})}$

Step 11.2.4.33

Combine $3$ and $\frac{5.562}{4}$ .

$\frac{1}{\frac{3 \cdot 5.562}{4}}$

Step 11.2.4.34

Multiply $3$ by $5.562$ .

$\frac{1}{\frac{16.686}{4}}$

Step 11.2.4.35

Multiply by the reciprocal of the fraction to divide by $\frac{16.686}{4}$ .

$1 (\frac{4}{16.686})$

Step 11.2.4.36

Multiply $\frac{4}{16.686}$ by $1$ .

$\frac{4}{16.686}$

Step 12

Divide $4$ by $16.686$ .

$0.23972192$

Step 13