Calculus Examples

$\int_{4}^{7} 6 (t^{2} - 11 t + 30) d t$

Step 1

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

$6 \int_{4}^{7} t^{2} - 11 t + 30 d t$

Step 2

Split the single integral into multiple integrals.

$6 (\int_{4}^{7} t^{2} d t + \int_{4}^{7} - 11 t d t + \int_{4}^{7} 30 d t)$

Step 3

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

$6 (\frac{1}{3} t^{3}]_{4}^{7} + \int_{4}^{7} - 11 t d t + \int_{4}^{7} 30 d t)$

Step 4

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

$6 (\frac{t^{3}}{3}]_{4}^{7} + \int_{4}^{7} - 11 t d t + \int_{4}^{7} 30 d t)$

Step 5

Since $- 11$ is constant with respect to $t$ , move $- 11$ out of the integral.

$6 (\frac{t^{3}}{3}]_{4}^{7} - 11 \int_{4}^{7} t d t + \int_{4}^{7} 30 d t)$

Step 6

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

$6 (\frac{t^{3}}{3}]_{4}^{7} - 11 (\frac{1}{2} t^{2}]_{4}^{7}) + \int_{4}^{7} 30 d t)$

Step 7

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

$6 (\frac{t^{3}}{3}]_{4}^{7} - 11 (\frac{t^{2}}{2}]_{4}^{7}) + \int_{4}^{7} 30 d t)$

Step 8

Apply the constant rule.

$6 (\frac{t^{3}}{3}]_{4}^{7} - 11 (\frac{t^{2}}{2}]_{4}^{7}) + 30 t]_{4}^{7})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{t^{3}}{3}]_{4}^{7}$ and $30 t]_{4}^{7}$ .

$6 (\frac{t^{3}}{3} + 30 t]_{4}^{7} - 11 (\frac{t^{2}}{2}]_{4}^{7}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{t^{3}}{3} + 30 t$ at $7$ and at $4$ .

$6 ((\frac{7^{3}}{3} + 30 \cdot 7) - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 (\frac{t^{2}}{2}]_{4}^{7}))$

Step 9.2.2

Evaluate $\frac{t^{2}}{2}$ at $7$ and at $4$ .

$6 ((\frac{7^{3}}{3} + 30 \cdot 7) - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Raise $7$ to the power of $3$ .

$6 (\frac{343}{3} + 30 \cdot 7 - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.2

Multiply $30$ by $7$ .

$6 (\frac{343}{3} + 210 - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.3

To write $210$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 (\frac{343}{3} + 210 \cdot \frac{3}{3} - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.4

Combine $210$ and $\frac{3}{3}$ .

$6 (\frac{343}{3} + \frac{210 \cdot 3}{3} - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.5

Combine the numerators over the common denominator.

$6 (\frac{343 + 210 \cdot 3}{3} - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.6

Simplify the numerator.

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Step 9.2.3.6.1

Multiply $210$ by $3$ .

$6 (\frac{343 + 630}{3} - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.6.2

Add $343$ and $630$ .

$6 (\frac{973}{3} - (\frac{4^{3}}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.7

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

$6 (\frac{973}{3} - (\frac{64}{3} + 30 \cdot 4) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.8

Multiply $30$ by $4$ .

$6 (\frac{973}{3} - (\frac{64}{3} + 120) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.9

To write $120$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 (\frac{973}{3} - (\frac{64}{3} + 120 \cdot \frac{3}{3}) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.10

Combine $120$ and $\frac{3}{3}$ .

$6 (\frac{973}{3} - (\frac{64}{3} + \frac{120 \cdot 3}{3}) - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.11

Combine the numerators over the common denominator.

$6 (\frac{973}{3} - \frac{64 + 120 \cdot 3}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.12

Simplify the numerator.

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Step 9.2.3.12.1

Multiply $120$ by $3$ .

$6 (\frac{973}{3} - \frac{64 + 360}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.12.2

Add $64$ and $360$ .

$6 (\frac{973}{3} - \frac{424}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.13

Combine the numerators over the common denominator.

$6 (\frac{973 - 424}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.14

Subtract $424$ from $973$ .

$6 (\frac{549}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.15

Cancel the common factor of $549$ and $3$ .

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Step 9.2.3.15.1

Factor $3$ out of $549$ .

$6 (\frac{3 \cdot 183}{3} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.15.2

Cancel the common factors.

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Step 9.2.3.15.2.1

Factor $3$ out of $3$ .

$6 (\frac{3 \cdot 183}{3 (1)} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.15.2.2

Cancel the common factor.

$6 (\frac{3 \cdot 183}{3 \cdot 1} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.15.2.3

Rewrite the expression.

$6 (\frac{183}{1} - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.15.2.4

Divide $183$ by $1$ .

$6 (183 - 11 ((\frac{7^{2}}{2}) - \frac{4^{2}}{2}))$

Step 9.2.3.16

Raise $7$ to the power of $2$ .

$6 (183 - 11 (\frac{49}{2} - \frac{4^{2}}{2}))$

Step 9.2.3.17

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

$6 (183 - 11 (\frac{49}{2} - \frac{16}{2}))$

Step 9.2.3.18

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

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Step 9.2.3.18.1

Factor $2$ out of $16$ .

$6 (183 - 11 (\frac{49}{2} - \frac{2 \cdot 8}{2}))$

Step 9.2.3.18.2

Cancel the common factors.

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Step 9.2.3.18.2.1

Factor $2$ out of $2$ .

$6 (183 - 11 (\frac{49}{2} - \frac{2 \cdot 8}{2 (1)}))$

Step 9.2.3.18.2.2

Cancel the common factor.

$6 (183 - 11 (\frac{49}{2} - \frac{2 \cdot 8}{2 \cdot 1}))$

Step 9.2.3.18.2.3

Rewrite the expression.

$6 (183 - 11 (\frac{49}{2} - \frac{8}{1}))$

Step 9.2.3.18.2.4

Divide $8$ by $1$ .

$6 (183 - 11 (\frac{49}{2} - 1 \cdot 8))$

Step 9.2.3.19

Multiply $- 1$ by $8$ .

$6 (183 - 11 (\frac{49}{2} - 8))$

Step 9.2.3.20

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

$6 (183 - 11 (\frac{49}{2} - 8 \cdot \frac{2}{2}))$

Step 9.2.3.21

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

$6 (183 - 11 (\frac{49}{2} + \frac{- 8 \cdot 2}{2}))$

Step 9.2.3.22

Combine the numerators over the common denominator.

$6 (183 - 11 \frac{49 - 8 \cdot 2}{2})$

Step 9.2.3.23

Simplify the numerator.

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Step 9.2.3.23.1

Multiply $- 8$ by $2$ .

$6 (183 - 11 \frac{49 - 16}{2})$

Step 9.2.3.23.2

Subtract $16$ from $49$ .

$6 (183 - 11 (\frac{33}{2}))$

Step 9.2.3.24

Combine $- 11$ and $\frac{33}{2}$ .

$6 (183 + \frac{- 11 \cdot 33}{2})$

Step 9.2.3.25

Multiply $- 11$ by $33$ .

$6 (183 + \frac{- 363}{2})$

Step 9.2.3.26

Move the negative in front of the fraction.

$6 (183 - \frac{363}{2})$

Step 9.2.3.27

To write $183$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 (183 \cdot \frac{2}{2} - \frac{363}{2})$

Step 9.2.3.28

Combine $183$ and $\frac{2}{2}$ .

$6 (\frac{183 \cdot 2}{2} - \frac{363}{2})$

Step 9.2.3.29

Combine the numerators over the common denominator.

$6 \frac{183 \cdot 2 - 363}{2}$

Step 9.2.3.30

Simplify the numerator.

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Step 9.2.3.30.1

Multiply $183$ by $2$ .

$6 \frac{366 - 363}{2}$

Step 9.2.3.30.2

Subtract $363$ from $366$ .

$6 (\frac{3}{2})$

Step 9.2.3.31

Combine $6$ and $\frac{3}{2}$ .

$\frac{6 \cdot 3}{2}$

Step 9.2.3.32

Multiply $6$ by $3$ .

$\frac{18}{2}$

Step 9.2.3.33

Cancel the common factor of $18$ and $2$ .

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Step 9.2.3.33.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9}{2}$

Step 9.2.3.33.2

Cancel the common factors.

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Step 9.2.3.33.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 9}{2 (1)}$

Step 9.2.3.33.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 1}$

Step 9.2.3.33.2.3

Rewrite the expression.

$\frac{9}{1}$

Step 9.2.3.33.2.4

Divide $9$ by $1$ .

$9$

Step 10