Calculus Examples

$\int_{- 1}^{1} \sqrt[3]{t} - 2 d t$

Step 1

Split the single integral into multiple integrals.

$\int_{- 1}^{1} \sqrt[3]{t} d t + \int_{- 1}^{1} - 2 d t$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t}$ as $t^{\frac{1}{3}}$ .

$\int_{- 1}^{1} t^{\frac{1}{3}} d t + \int_{- 1}^{1} - 2 d t$

Step 3

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

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{1} + \int_{- 1}^{1} - 2 d t$

Step 4

Apply the constant rule.

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{1} + - 2 t]_{- 1}^{1}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{1}$ and $- 2 t]_{- 1}^{1}$ .

$\frac{3}{4} t^{\frac{4}{3}} - 2 t]_{- 1}^{1}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{3}{4} t^{\frac{4}{3}} - 2 t$ at $1$ and at $- 1$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

One to any power is one.

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

Step 5.2.2.2

Multiply $\frac{3}{4}$ by $1$ .

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

Step 5.2.2.3

Multiply $- 2$ by $1$ .

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

Step 5.2.2.4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 5.2.2.5

Combine $- 2$ and $\frac{4}{4}$ .

$\frac{3}{4} + \frac{- 2 \cdot 4}{4} - (\frac{3}{4} {(- 1)}^{\frac{4}{3}} - 2 \cdot - 1)$

Step 5.2.2.6

Combine the numerators over the common denominator.

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

Step 5.2.2.7

Simplify the numerator.

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Step 5.2.2.7.1

Multiply $- 2$ by $4$ .

$\frac{3 - 8}{4} - (\frac{3}{4} {(- 1)}^{\frac{4}{3}} - 2 \cdot - 1)$

Step 5.2.2.7.2

Subtract $8$ from $3$ .

$\frac{- 5}{4} - (\frac{3}{4} {(- 1)}^{\frac{4}{3}} - 2 \cdot - 1)$

Step 5.2.2.8

Move the negative in front of the fraction.

$- \frac{5}{4} - (\frac{3}{4} {(- 1)}^{\frac{4}{3}} - 2 \cdot - 1)$

Step 5.2.2.9

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- \frac{5}{4} - (\frac{3}{4} {({(- 1)}^{3})}^{\frac{4}{3}} - 2 \cdot - 1)$

Step 5.2.2.10

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{5}{4} - (\frac{3}{4} {(- 1)}^{3 (\frac{4}{3})} - 2 \cdot - 1)$

Step 5.2.2.11

Cancel the common factor of $3$ .

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Step 5.2.2.11.1

Cancel the common factor.

$- \frac{5}{4} - (\frac{3}{4} {(- 1)}^{3 (\frac{4}{3})} - 2 \cdot - 1)$

Step 5.2.2.11.2

Rewrite the expression.

$- \frac{5}{4} - (\frac{3}{4} {(- 1)}^{4} - 2 \cdot - 1)$

Step 5.2.2.12

Raise $- 1$ to the power of $4$ .

$- \frac{5}{4} - (\frac{3}{4} \cdot 1 - 2 \cdot - 1)$

Step 5.2.2.13

Multiply $\frac{3}{4}$ by $1$ .

$- \frac{5}{4} - (\frac{3}{4} - 2 \cdot - 1)$

Step 5.2.2.14

Multiply $- 2$ by $- 1$ .

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

Step 5.2.2.15

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 5.2.2.16

Combine $2$ and $\frac{4}{4}$ .

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

Step 5.2.2.17

Combine the numerators over the common denominator.

$- \frac{5}{4} - \frac{3 + 2 \cdot 4}{4}$

Step 5.2.2.18

Simplify the numerator.

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Step 5.2.2.18.1

Multiply $2$ by $4$ .

$- \frac{5}{4} - \frac{3 + 8}{4}$

Step 5.2.2.18.2

Add $3$ and $8$ .

$- \frac{5}{4} - \frac{11}{4}$

Step 5.2.2.19

Combine the numerators over the common denominator.

$\frac{- 5 - 11}{4}$

Step 5.2.2.20

Subtract $11$ from $- 5$ .

$\frac{- 16}{4}$

Step 5.2.2.21

Cancel the common factor of $- 16$ and $4$ .

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Step 5.2.2.21.1

Factor $4$ out of $- 16$ .

$\frac{4 \cdot - 4}{4}$

Step 5.2.2.21.2

Cancel the common factors.

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Step 5.2.2.21.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 4}{4 (1)}$

Step 5.2.2.21.2.2

Cancel the common factor.

$\frac{4 \cdot - 4}{4 \cdot 1}$

Step 5.2.2.21.2.3

Rewrite the expression.

$\frac{- 4}{1}$

Step 5.2.2.21.2.4

Divide $- 4$ by $1$ .

$- 4$

Step 6