Calculus Examples

$\int_{0}^{1} (u + 5) (u - 6) d u$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int_{0}^{1} u (u - 6) + 5 (u - 6) d u$

Step 1.2

Apply the distributive property.

$\int_{0}^{1} u \cdot u + u \cdot - 6 + 5 (u - 6) d u$

Step 1.3

Apply the distributive property.

$\int_{0}^{1} u \cdot u + u \cdot - 6 + 5 u + 5 \cdot - 6 d u$

Step 1.4

Reorder $u$ and $- 6$ .

$\int_{0}^{1} u \cdot u - 6 \cdot u + 5 u + 5 \cdot - 6 d u$

Step 1.5

Raise $u$ to the power of $1$ .

$\int_{0}^{1} u^{1} u - 6 u + 5 u + 5 \cdot - 6 d u$

Step 1.6

Raise $u$ to the power of $1$ .

$\int_{0}^{1} u^{1} u^{1} - 6 u + 5 u + 5 \cdot - 6 d u$

Step 1.7

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

$\int_{0}^{1} u^{1 + 1} - 6 u + 5 u + 5 \cdot - 6 d u$

Step 1.8

Add $1$ and $1$ .

$\int_{0}^{1} u^{2} - 6 u + 5 u + 5 \cdot - 6 d u$

Step 1.9

Multiply $5$ by $- 6$ .

$\int_{0}^{1} u^{2} - 6 u + 5 u - 30 d u$

Step 1.10

Add $- 6 u$ and $5 u$ .

$\int_{0}^{1} u^{2} - u - 30 d u$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} u^{2} d u + \int_{0}^{1} - u d u + \int_{0}^{1} - 30 d u$

Step 3

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

$\frac{1}{3} u^{3}]_{0}^{1} + \int_{0}^{1} - u d u + \int_{0}^{1} - 30 d u$

Step 4

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

$\frac{1}{3} u^{3}]_{0}^{1} - \int_{0}^{1} u d u + \int_{0}^{1} - 30 d u$

Step 5

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

$\frac{1}{3} u^{3}]_{0}^{1} - (\frac{1}{2} u^{2}]_{0}^{1}) + \int_{0}^{1} - 30 d u$

Step 6

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

$\frac{1}{3} u^{3}]_{0}^{1} - (\frac{u^{2}}{2}]_{0}^{1}) + \int_{0}^{1} - 30 d u$

Step 7

Apply the constant rule.

$\frac{1}{3} u^{3}]_{0}^{1} - (\frac{u^{2}}{2}]_{0}^{1}) + - 30 u]_{0}^{1}$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3} u^{3}]_{0}^{1}$ and $- 30 u]_{0}^{1}$ .

$\frac{1}{3} u^{3} - 30 u]_{0}^{1} - (\frac{u^{2}}{2}]_{0}^{1})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{1}{3} u^{3} - 30 u$ at $1$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3} - 30 \cdot 1) - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - (\frac{u^{2}}{2}]_{0}^{1})$

Step 8.2.2

Evaluate $\frac{u^{2}}{2}$ at $1$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3} - 30 \cdot 1) - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

One to any power is one.

$\frac{1}{3} \cdot 1 - 30 \cdot 1 - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2

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

$\frac{1}{3} - 30 \cdot 1 - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.3

Multiply $- 30$ by $1$ .

$\frac{1}{3} - 30 - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4

To write $- 30$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} - 30 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.5

Combine $- 30$ and $\frac{3}{3}$ .

$\frac{1}{3} + \frac{- 30 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.6

Combine the numerators over the common denominator.

$\frac{1 - 30 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7

Simplify the numerator.

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Step 8.2.3.7.1

Multiply $- 30$ by $3$ .

$\frac{1 - 90}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7.2

Subtract $90$ from $1$ .

$\frac{- 89}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.8

Move the negative in front of the fraction.

$- \frac{89}{3} - (\frac{1}{3} \cdot 0^{3} - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.9

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

$- \frac{89}{3} - (\frac{1}{3} \cdot 0 - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.10

Multiply $\frac{1}{3}$ by $0$ .

$- \frac{89}{3} - (0 - 30 \cdot 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.11

Multiply $- 30$ by $0$ .

$- \frac{89}{3} - (0 + 0) - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.12

Add $0$ and $0$ .

$- \frac{89}{3} - 0 - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.13

Multiply $- 1$ by $0$ .

$- \frac{89}{3} + 0 - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.14

Add $- \frac{89}{3}$ and $0$ .

$- \frac{89}{3} - ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.15

One to any power is one.

$- \frac{89}{3} - (\frac{1}{2} - \frac{0^{2}}{2})$

Step 8.2.3.16

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

$- \frac{89}{3} - (\frac{1}{2} - \frac{0}{2})$

Step 8.2.3.17

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

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Step 8.2.3.17.1

Factor $2$ out of $0$ .

$- \frac{89}{3} - (\frac{1}{2} - \frac{2 (0)}{2})$

Step 8.2.3.17.2

Cancel the common factors.

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Step 8.2.3.17.2.1

Factor $2$ out of $2$ .

$- \frac{89}{3} - (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.17.2.2

Cancel the common factor.

$- \frac{89}{3} - (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.17.2.3

Rewrite the expression.

$- \frac{89}{3} - (\frac{1}{2} - \frac{0}{1})$

Step 8.2.3.17.2.4

Divide $0$ by $1$ .

$- \frac{89}{3} - (\frac{1}{2} - 0)$

Step 8.2.3.18

Multiply $- 1$ by $0$ .

$- \frac{89}{3} - (\frac{1}{2} + 0)$

Step 8.2.3.19

Add $\frac{1}{2}$ and $0$ .

$- \frac{89}{3} - \frac{1}{2}$

Step 8.2.3.20

To write $- \frac{89}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{89}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 8.2.3.21

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{89}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.2.3.22

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

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Step 8.2.3.22.1

Multiply $\frac{89}{3}$ by $\frac{2}{2}$ .

$- \frac{89 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.2.3.22.2

Multiply $3$ by $2$ .

$- \frac{89 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.2.3.22.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$- \frac{89 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 8.2.3.22.4

Multiply $2$ by $3$ .

$- \frac{89 \cdot 2}{6} - \frac{3}{6}$

Step 8.2.3.23

Combine the numerators over the common denominator.

$\frac{- 89 \cdot 2 - 3}{6}$

Step 8.2.3.24

Simplify the numerator.

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Step 8.2.3.24.1

Multiply $- 89$ by $2$ .

$\frac{- 178 - 3}{6}$

Step 8.2.3.24.2

Subtract $3$ from $- 178$ .

$\frac{- 181}{6}$

Step 8.2.3.25

Move the negative in front of the fraction.

$- \frac{181}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{181}{6}$

Decimal Form:

$- 30.1\overline{6}$

Mixed Number Form:

$- 30 \frac{1}{6}$

Step 10