Calculus Examples

$\int_{2}^{5} 2 + 2 t + 3 t^{2} d t$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} 2 d t + \int_{2}^{5} 2 t d t + \int_{2}^{5} 3 t^{2} d t$

Step 2

Apply the constant rule.

$2 t]_{2}^{5} + \int_{2}^{5} 2 t d t + \int_{2}^{5} 3 t^{2} d t$

Step 3

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$2 t]_{2}^{5} + 2 \int_{2}^{5} t d t + \int_{2}^{5} 3 t^{2} d t$

Step 4

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

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

Step 5

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

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

Step 6

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $2 t$ at $5$ and at $2$ .

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

Simplify.

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Step 8.2.4.1

Multiply $2$ by $5$ .

$10 - 2 \cdot 2 + 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.2

Multiply $- 2$ by $2$ .

$10 - 4 + 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.3

Subtract $4$ from $10$ .

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

Step 8.2.4.4

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

$6 + 2 (\frac{25}{2} - \frac{2^{2}}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.5

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

$6 + 2 (\frac{25}{2} - \frac{4}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.6

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

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Step 8.2.4.6.1

Factor $2$ out of $4$ .

$6 + 2 (\frac{25}{2} - \frac{2 \cdot 2}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.6.2

Cancel the common factors.

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Step 8.2.4.6.2.1

Factor $2$ out of $2$ .

$6 + 2 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.6.2.2

Cancel the common factor.

$6 + 2 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.6.2.3

Rewrite the expression.

$6 + 2 (\frac{25}{2} - \frac{2}{1}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.6.2.4

Divide $2$ by $1$ .

$6 + 2 (\frac{25}{2} - 1 \cdot 2) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.7

Multiply $- 1$ by $2$ .

$6 + 2 (\frac{25}{2} - 2) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.8

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

$6 + 2 (\frac{25}{2} - 2 \cdot \frac{2}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.9

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

$6 + 2 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.10

Combine the numerators over the common denominator.

$6 + 2 \frac{25 - 2 \cdot 2}{2} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.11

Simplify the numerator.

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Step 8.2.4.11.1

Multiply $- 2$ by $2$ .

$6 + 2 \frac{25 - 4}{2} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.11.2

Subtract $4$ from $25$ .

$6 + 2 (\frac{21}{2}) + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.12

Combine $2$ and $\frac{21}{2}$ .

$6 + \frac{2 \cdot 21}{2} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.13

Multiply $2$ by $21$ .

$6 + \frac{42}{2} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.14

Cancel the common factor of $42$ and $2$ .

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Step 8.2.4.14.1

Factor $2$ out of $42$ .

$6 + \frac{2 \cdot 21}{2} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.14.2

Cancel the common factors.

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Step 8.2.4.14.2.1

Factor $2$ out of $2$ .

$6 + \frac{2 \cdot 21}{2 (1)} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.14.2.2

Cancel the common factor.

$6 + \frac{2 \cdot 21}{2 \cdot 1} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.14.2.3

Rewrite the expression.

$6 + \frac{21}{1} + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.14.2.4

Divide $21$ by $1$ .

$6 + 21 + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.15

Add $6$ and $21$ .

$27 + 3 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3})$

Step 8.2.4.16

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

$27 + 3 (\frac{125}{3} - \frac{2^{3}}{3})$

Step 8.2.4.17

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

$27 + 3 (\frac{125}{3} - \frac{8}{3})$

Step 8.2.4.18

Combine the numerators over the common denominator.

$27 + 3 \frac{125 - 8}{3}$

Step 8.2.4.19

Subtract $8$ from $125$ .

$27 + 3 (\frac{117}{3})$

Step 8.2.4.20

Cancel the common factor of $117$ and $3$ .

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Step 8.2.4.20.1

Factor $3$ out of $117$ .

$27 + 3 \frac{3 \cdot 39}{3}$

Step 8.2.4.20.2

Cancel the common factors.

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Step 8.2.4.20.2.1

Factor $3$ out of $3$ .

$27 + 3 \frac{3 \cdot 39}{3 (1)}$

Step 8.2.4.20.2.2

Cancel the common factor.

$27 + 3 \frac{3 \cdot 39}{3 \cdot 1}$

Step 8.2.4.20.2.3

Rewrite the expression.

$27 + 3 (\frac{39}{1})$

Step 8.2.4.20.2.4

Divide $39$ by $1$ .

$27 + 3 \cdot 39$

Step 8.2.4.21

Multiply $3$ by $39$ .

$27 + 117$

Step 8.2.4.22

Add $27$ and $117$ .

$144$

Step 9