Calculus Examples

$\int_{2}^{3} (t^{2} - 9) (4 - t^{2}) d t$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int_{2}^{3} t^{2} (4 - t^{2}) - 9 (4 - t^{2}) d t$

Step 1.2

Apply the distributive property.

$\int_{2}^{3} t^{2} \cdot 4 + t^{2} (- t^{2}) - 9 (4 - t^{2}) d t$

Step 1.3

Apply the distributive property.

$\int_{2}^{3} t^{2} \cdot 4 + t^{2} (- t^{2}) - 9 \cdot 4 - 9 (- t^{2}) d t$

Step 1.4

Reorder $t^{2}$ and $4$ .

$\int_{2}^{3} 4 \cdot t^{2} + t^{2} (- t^{2}) - 9 \cdot 4 - 9 (- t^{2}) d t$

Step 1.5

Reorder $t^{2}$ and $- 1$ .

$\int_{2}^{3} 4 \cdot t^{2} - 1 \cdot t^{2} t^{2} - 9 \cdot 4 - 9 (- t^{2}) d t$

Step 1.6

Factor out negative.

$\int_{2}^{3} 4 t^{2} - (t^{2} t^{2}) - 9 \cdot 4 - 9 \cdot - 1 t^{2} d t$

Step 1.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{2}^{3} 4 t^{2} - t^{2 + 2} - 9 \cdot 4 - 9 \cdot - 1 t^{2} d t$

Step 1.8

Add $2$ and $2$ .

$\int_{2}^{3} 4 t^{2} - t^{4} - 9 \cdot 4 - 9 \cdot - 1 t^{2} d t$

Step 1.9

Multiply $- 9$ by $4$ .

$\int_{2}^{3} 4 t^{2} - t^{4} - 36 - 9 \cdot - 1 t^{2} d t$

Step 1.10

Multiply $- 9$ by $- 1$ .

$\int_{2}^{3} 4 t^{2} - t^{4} - 36 + 9 t^{2} d t$

Step 1.11

Reorder $4 t^{2}$ and $- t^{4}$ .

$\int_{2}^{3} - t^{4} + 4 t^{2} - 36 + 9 t^{2} d t$

Step 1.12

Reorder $- 36$ and $9 t^{2}$ .

$\int_{2}^{3} - t^{4} + 4 t^{2} + 9 t^{2} - 36 d t$

Step 1.13

Move $4 t^{2}$ .

$\int_{2}^{3} - t^{4} + 9 t^{2} + 4 t^{2} - 36 d t$

Step 1.14

Add $9 t^{2}$ and $4 t^{2}$ .

$\int_{2}^{3} - t^{4} + 13 t^{2} - 36 d t$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{3} - t^{4} d t + \int_{2}^{3} 13 t^{2} d t + \int_{2}^{3} - 36 d t$

Step 3

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

$- \int_{2}^{3} t^{4} d t + \int_{2}^{3} 13 t^{2} d t + \int_{2}^{3} - 36 d t$

Step 4

By the Power Rule, the integral of $t^{4}$ with respect to $t$ is $\frac{1}{5} t^{5}$ .

$- (\frac{1}{5} t^{5}]_{2}^{3}) + \int_{2}^{3} 13 t^{2} d t + \int_{2}^{3} - 36 d t$

Step 5

Combine $\frac{1}{5}$ and $t^{5}$ .

$- (\frac{t^{5}}{5}]_{2}^{3}) + \int_{2}^{3} 13 t^{2} d t + \int_{2}^{3} - 36 d t$

Step 6

Since $13$ is constant with respect to $t$ , move $13$ out of the integral.

$- (\frac{t^{5}}{5}]_{2}^{3}) + 13 \int_{2}^{3} t^{2} d t + \int_{2}^{3} - 36 d t$

Step 7

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

$- (\frac{t^{5}}{5}]_{2}^{3}) + 13 (\frac{1}{3} t^{3}]_{2}^{3}) + \int_{2}^{3} - 36 d t$

Step 8

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

$- (\frac{t^{5}}{5}]_{2}^{3}) + 13 (\frac{t^{3}}{3}]_{2}^{3}) + \int_{2}^{3} - 36 d t$

Step 9

Apply the constant rule.

$- (\frac{t^{5}}{5}]_{2}^{3}) + 13 (\frac{t^{3}}{3}]_{2}^{3}) + - 36 t]_{2}^{3}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{t^{5}}{5}$ at $3$ and at $2$ .

$- ((\frac{3^{5}}{5}) - \frac{2^{5}}{5}) + 13 (\frac{t^{3}}{3}]_{2}^{3}) + - 36 t]_{2}^{3}$

Step 10.2

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

$- (\frac{3^{5}}{5} - \frac{2^{5}}{5}) + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + - 36 t]_{2}^{3}$

Step 10.3

Evaluate $- 36 t$ at $3$ and at $2$ .

$- (\frac{3^{5}}{5} - \frac{2^{5}}{5}) + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4

Simplify.

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Step 10.4.1

Raise $3$ to the power of $5$ .

$- (\frac{243}{5} - \frac{2^{5}}{5}) + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.2

Raise $2$ to the power of $5$ .

$- (\frac{243}{5} - \frac{32}{5}) + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.3

Combine the numerators over the common denominator.

$- \frac{243 - 32}{5} + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.4

Subtract $32$ from $243$ .

$- \frac{211}{5} + 13 (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.5

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

$- \frac{211}{5} + 13 (\frac{27}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.6

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

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Step 10.4.6.1

Factor $3$ out of $27$ .

$- \frac{211}{5} + 13 (\frac{3 \cdot 9}{3} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.6.2

Cancel the common factors.

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Step 10.4.6.2.1

Factor $3$ out of $3$ .

$- \frac{211}{5} + 13 (\frac{3 \cdot 9}{3 (1)} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.6.2.2

Cancel the common factor.

$- \frac{211}{5} + 13 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.6.2.3

Rewrite the expression.

$- \frac{211}{5} + 13 (\frac{9}{1} - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.6.2.4

Divide $9$ by $1$ .

$- \frac{211}{5} + 13 (9 - \frac{2^{3}}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.7

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

$- \frac{211}{5} + 13 (9 - \frac{8}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.8

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

$- \frac{211}{5} + 13 (9 \cdot \frac{3}{3} - \frac{8}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.9

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

$- \frac{211}{5} + 13 (\frac{9 \cdot 3}{3} - \frac{8}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.10

Combine the numerators over the common denominator.

$- \frac{211}{5} + 13 \frac{9 \cdot 3 - 8}{3} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.11

Simplify the numerator.

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Step 10.4.11.1

Multiply $9$ by $3$ .

$- \frac{211}{5} + 13 \frac{27 - 8}{3} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.11.2

Subtract $8$ from $27$ .

$- \frac{211}{5} + 13 (\frac{19}{3}) + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.12

Combine $13$ and $\frac{19}{3}$ .

$- \frac{211}{5} + \frac{13 \cdot 19}{3} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.13

Multiply $13$ by $19$ .

$- \frac{211}{5} + \frac{247}{3} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.14

To write $- \frac{211}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{211}{5} \cdot \frac{3}{3} + \frac{247}{3} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.15

To write $\frac{247}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{211}{5} \cdot \frac{3}{3} + \frac{247}{3} \cdot \frac{5}{5} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.16

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.4.16.1

Multiply $\frac{211}{5}$ by $\frac{3}{3}$ .

$- \frac{211 \cdot 3}{5 \cdot 3} + \frac{247}{3} \cdot \frac{5}{5} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.16.2

Multiply $5$ by $3$ .

$- \frac{211 \cdot 3}{15} + \frac{247}{3} \cdot \frac{5}{5} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.16.3

Multiply $\frac{247}{3}$ by $\frac{5}{5}$ .

$- \frac{211 \cdot 3}{15} + \frac{247 \cdot 5}{3 \cdot 5} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.16.4

Multiply $3$ by $5$ .

$- \frac{211 \cdot 3}{15} + \frac{247 \cdot 5}{15} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.17

Combine the numerators over the common denominator.

$\frac{- 211 \cdot 3 + 247 \cdot 5}{15} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.18

Simplify the numerator.

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Step 10.4.18.1

Multiply $- 211$ by $3$ .

$\frac{- 633 + 247 \cdot 5}{15} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.18.2

Multiply $247$ by $5$ .

$\frac{- 633 + 1235}{15} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.18.3

Add $- 633$ and $1235$ .

$\frac{602}{15} + (- 36 \cdot 3) + 36 \cdot 2$

Step 10.4.19

Multiply $- 36$ by $3$ .

$\frac{602}{15} - 108 + 36 \cdot 2$

Step 10.4.20

Multiply $36$ by $2$ .

$\frac{602}{15} - 108 + 72$

Step 10.4.21

Add $- 108$ and $72$ .

$\frac{602}{15} - 36$

Step 10.4.22

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{602}{15} - 36 \cdot \frac{15}{15}$

Step 10.4.23

Combine $- 36$ and $\frac{15}{15}$ .

$\frac{602}{15} + \frac{- 36 \cdot 15}{15}$

Step 10.4.24

Combine the numerators over the common denominator.

$\frac{602 - 36 \cdot 15}{15}$

Step 10.4.25

Simplify the numerator.

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Step 10.4.25.1

Multiply $- 36$ by $15$ .

$\frac{602 - 540}{15}$

Step 10.4.25.2

Subtract $540$ from $602$ .

$\frac{62}{15}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{62}{15}$

Decimal Form:

$4.1\overline{3}$

Mixed Number Form:

$4 \frac{2}{15}$

Step 12