Calculus Examples

$\int_{2}^{4} x^{\frac{1}{3}} (1 - 2 x) d x$

Step 1

Expand $x^{\frac{1}{3}} (1 - 2 x)$ .

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Step 1.1

Apply the distributive property.

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

Step 1.2

Reorder $x^{\frac{1}{3}}$ and $1$ .

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

Step 1.3

Reorder $x^{\frac{1}{3}}$ and $- 2$ .

$\int_{2}^{4} 1 \cdot x^{\frac{1}{3}} - 2 \cdot x^{\frac{1}{3}} x d x$

Step 1.4

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

$\int_{2}^{4} x^{\frac{1}{3}} - 2 x^{\frac{1}{3}} x d x$

Step 1.5

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

$\int_{2}^{4} x^{\frac{1}{3}} - 2 (x^{\frac{1}{3}} x^{1}) d x$

Step 1.6

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

$\int_{2}^{4} x^{\frac{1}{3}} - 2 x^{\frac{1}{3} + 1} d x$

Step 1.7

Write $1$ as a fraction with a common denominator.

$\int_{2}^{4} x^{\frac{1}{3}} - 2 x^{\frac{1}{3} + \frac{3}{3}} d x$

Step 1.8

Combine the numerators over the common denominator.

$\int_{2}^{4} x^{\frac{1}{3}} - 2 x^{\frac{1 + 3}{3}} d x$

Step 1.9

Add $1$ and $3$ .

$\int_{2}^{4} x^{\frac{1}{3}} - 2 x^{\frac{4}{3}} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{4} x^{\frac{1}{3}} d x + \int_{2}^{4} - 2 x^{\frac{4}{3}} d x$

Step 3

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

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

Step 4

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

$\frac{3}{4} x^{\frac{4}{3}}]_{2}^{4} - 2 \int_{2}^{4} x^{\frac{4}{3}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$\frac{3}{4} x^{\frac{4}{3}}]_{2}^{4} - 2 (\frac{3}{7} x^{\frac{7}{3}}]_{2}^{4})$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{3}{7}$ and $x^{\frac{7}{3}}$ .

$\frac{3}{4} x^{\frac{4}{3}}]_{2}^{4} - 2 (\frac{3 x^{\frac{7}{3}}}{7}]_{2}^{4})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}}$ at $4$ and at $2$ .

$(\frac{3}{4} \cdot 4^{\frac{4}{3}}) - \frac{3}{4} \cdot 2^{\frac{4}{3}} - 2 (\frac{3 x^{\frac{7}{3}}}{7}]_{2}^{4})$

Step 6.2.2

Evaluate $\frac{3 x^{\frac{7}{3}}}{7}$ at $4$ and at $2$ .

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Combine $\frac{3}{4}$ and $4^{\frac{4}{3}}$ .

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

Step 6.2.3.2

Move $4^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 6.2.3.3

Multiply $4^{\frac{4}{3}}$ by $4^{- 1}$ by adding the exponents.

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Step 6.2.3.3.1

Move $4^{- 1}$ .

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

Step 6.2.3.3.2

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

$3 \cdot 4^{- 1 + \frac{4}{3}} - \frac{3}{4} \cdot 2^{\frac{4}{3}} - 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$

Step 6.2.3.3.3

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

$3 \cdot 4^{- 1 \cdot \frac{3}{3} + \frac{4}{3}} - \frac{3}{4} \cdot 2^{\frac{4}{3}} - 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$

Step 6.2.3.3.4

Combine $- 1$ and $\frac{3}{3}$ .

$3 \cdot 4^{\frac{- 1 \cdot 3}{3} + \frac{4}{3}} - \frac{3}{4} \cdot 2^{\frac{4}{3}} - 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$

Step 6.2.3.3.5

Combine the numerators over the common denominator.

$3 \cdot 4^{\frac{- 1 \cdot 3 + 4}{3}} - \frac{3}{4} \cdot 2^{\frac{4}{3}} - 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$

Step 6.2.3.3.6

Simplify the numerator.

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Step 6.2.3.3.6.1

Multiply $- 1$ by $3$ .

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

Step 6.2.3.3.6.2

Add $- 3$ and $4$ .

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

Step 6.2.3.4

Combine $2^{\frac{4}{3}}$ and $\frac{3}{4}$ .

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

Step 6.2.3.5

Move $3$ to the left of $2^{\frac{4}{3}}$ .

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

Step 6.2.3.6

To write $- 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.2.3.7

Combine $- 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})$ and $\frac{4}{4}$ .

$3 \cdot 4^{\frac{1}{3}} - \frac{3 \cdot 2^{\frac{4}{3}}}{4} + \frac{- 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7}) \cdot 4}{4}$

Step 6.2.3.8

Combine the numerators over the common denominator.

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} - 2 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7}) \cdot 4}{4}$

Step 6.2.3.9

Multiply $4$ by $- 2$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} - 8 (\frac{3 \cdot 4^{\frac{7}{3}}}{7} - \frac{3 \cdot 2^{\frac{7}{3}}}{7})}{4}$

Step 6.3

Simplify.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Simplify the numerator.

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Step 6.3.1.1.1

Apply the distributive property.

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} - 8 \frac{3 \cdot 4^{\frac{7}{3}}}{7} - 8 (- \frac{3 \cdot 2^{\frac{7}{3}}}{7})}{4}$

Step 6.3.1.1.2

Multiply $- 8 \frac{3 \cdot 4^{\frac{7}{3}}}{7}$ .

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Step 6.3.1.1.2.1

Combine $- 8$ and $\frac{3 \cdot 4^{\frac{7}{3}}}{7}$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} + \frac{- 8 (3 \cdot 4^{\frac{7}{3}})}{7} - 8 (- \frac{3 \cdot 2^{\frac{7}{3}}}{7})}{4}$

Step 6.3.1.1.2.2

Multiply $3$ by $- 8$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} + \frac{- 24 \cdot 4^{\frac{7}{3}}}{7} - 8 (- \frac{3 \cdot 2^{\frac{7}{3}}}{7})}{4}$

Step 6.3.1.1.3

Multiply $- 8 (- \frac{3 \cdot 2^{\frac{7}{3}}}{7})$ .

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Step 6.3.1.1.3.1

Multiply $- 1$ by $- 8$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} + \frac{- 24 \cdot 4^{\frac{7}{3}}}{7} + 8 \frac{3 \cdot 2^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.3.2

Combine $8$ and $\frac{3 \cdot 2^{\frac{7}{3}}}{7}$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} + \frac{- 24 \cdot 4^{\frac{7}{3}}}{7} + \frac{8 (3 \cdot 2^{\frac{7}{3}})}{7}}{4}$

Step 6.3.1.1.3.3

Multiply $3$ by $8$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} + \frac{- 24 \cdot 4^{\frac{7}{3}}}{7} + \frac{24 \cdot 2^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.4

Move the negative in front of the fraction.

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} - \frac{24 \cdot 4^{\frac{7}{3}}}{7} + \frac{24 \cdot 2^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.5

To write $- 3 \cdot 2^{\frac{4}{3}}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 3 \cdot 2^{\frac{4}{3}} \cdot \frac{7}{7} + \frac{24 \cdot 2^{\frac{7}{3}}}{7} - \frac{24 \cdot 4^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.6

Combine $- 3 \cdot 2^{\frac{4}{3}}$ and $\frac{7}{7}$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{\frac{- 3 \cdot 2^{\frac{4}{3}} \cdot 7}{7} + \frac{24 \cdot 2^{\frac{7}{3}}}{7} - \frac{24 \cdot 4^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.7

Combine the numerators over the common denominator.

$3 \cdot 4^{\frac{1}{3}} + \frac{\frac{- 3 \cdot 2^{\frac{4}{3}} \cdot 7 + 24 \cdot 2^{\frac{7}{3}}}{7} - \frac{24 \cdot 4^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.8

Combine the numerators over the common denominator.

$3 \cdot 4^{\frac{1}{3}} + \frac{\frac{- 3 \cdot 2^{\frac{4}{3}} \cdot 7 + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.1.9

Multiply $7$ by $- 3$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{\frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7}}{4}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$3 \cdot 4^{\frac{1}{3}} + \frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7} \cdot \frac{1}{4}$

Step 6.3.1.3

Multiply $\frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7} \cdot \frac{1}{4}$ .

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Step 6.3.1.3.1

Multiply $\frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7}$ by $\frac{1}{4}$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{7 \cdot 4}$

Step 6.3.1.3.2

Multiply $7$ by $4$ .

$3 \cdot 4^{\frac{1}{3}} + \frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Step 6.3.2

To write $3 \cdot 4^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{28}{28}$ .

$3 \cdot 4^{\frac{1}{3}} \cdot \frac{28}{28} + \frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Step 6.3.3

Combine $3 \cdot 4^{\frac{1}{3}}$ and $\frac{28}{28}$ .

$\frac{3 \cdot 4^{\frac{1}{3}} \cdot 28}{28} + \frac{- 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Step 6.3.4

Combine the numerators over the common denominator.

$\frac{3 \cdot 4^{\frac{1}{3}} \cdot 28 - 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Step 6.3.5

Multiply $28$ by $3$ .

$\frac{84 \cdot 4^{\frac{1}{3}} - 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{84 \cdot 4^{\frac{1}{3}} - 21 \cdot 2^{\frac{4}{3}} + 24 \cdot 2^{\frac{7}{3}} - 24 \cdot 4^{\frac{7}{3}}}{28}$

Decimal Form:

$- 14.57802067 \dots$

Step 8