Calculus Examples

$\int_{27}^{64} \sqrt[3]{x} - 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{27}^{64} \sqrt[3]{x} d x + \int_{27}^{64} - 3 d x$

Step 2

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

$\int_{27}^{64} x^{\frac{1}{3}} d x + \int_{27}^{64} - 3 d x$

Step 3

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

$\frac{3}{4} x^{\frac{4}{3}}]_{27}^{64} + \int_{27}^{64} - 3 d x$

Step 4

Apply the constant rule.

$\frac{3}{4} x^{\frac{4}{3}}]_{27}^{64} + - 3 x]_{27}^{64}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{3}{4} x^{\frac{4}{3}}]_{27}^{64}$ and $- 3 x]_{27}^{64}$ .

$\frac{3}{4} x^{\frac{4}{3}} - 3 x]_{27}^{64}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}} - 3 x$ at $64$ and at $27$ .

$(\frac{3}{4} \cdot 64^{\frac{4}{3}} - 3 \cdot 64) - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Rewrite $64$ as $4^{3}$ .

$\frac{3}{4} \cdot {(4^{3})}^{\frac{4}{3}} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.2

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

$\frac{3}{4} \cdot 4^{3 (\frac{4}{3})} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.3

Cancel the common factor of $3$ .

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{3}{4} \cdot 4^{3 (\frac{4}{3})} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.3.2

Rewrite the expression.

$\frac{3}{4} \cdot 4^{4} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.4

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

$\frac{3}{4} \cdot 256 - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.5

Combine $\frac{3}{4}$ and $256$ .

$\frac{3 \cdot 256}{4} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.6

Multiply $3$ by $256$ .

$\frac{768}{4} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.7

Cancel the common factor of $768$ and $4$ .

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Step 5.2.2.7.1

Factor $4$ out of $768$ .

$\frac{4 \cdot 192}{4} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.7.2

Cancel the common factors.

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Step 5.2.2.7.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 192}{4 (1)} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.7.2.2

Cancel the common factor.

$\frac{4 \cdot 192}{4 \cdot 1} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.7.2.3

Rewrite the expression.

$\frac{192}{1} - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.7.2.4

Divide $192$ by $1$ .

$192 - 3 \cdot 64 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.8

Multiply $- 3$ by $64$ .

$192 - 192 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.9

Subtract $192$ from $192$ .

$0 - (\frac{3}{4} \cdot 27^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.10

Rewrite $27$ as $3^{3}$ .

$0 - (\frac{3}{4} \cdot {(3^{3})}^{\frac{4}{3}} - 3 \cdot 27)$

Step 5.2.2.11

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

$0 - (\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - 3 \cdot 27)$

Step 5.2.2.12

Cancel the common factor of $3$ .

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Step 5.2.2.12.1

Cancel the common factor.

$0 - (\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - 3 \cdot 27)$

Step 5.2.2.12.2

Rewrite the expression.

$0 - (\frac{3}{4} \cdot 3^{4} - 3 \cdot 27)$

Step 5.2.2.13

Raise $3$ to the power of $4$ .

$0 - (\frac{3}{4} \cdot 81 - 3 \cdot 27)$

Step 5.2.2.14

Combine $\frac{3}{4}$ and $81$ .

$0 - (\frac{3 \cdot 81}{4} - 3 \cdot 27)$

Step 5.2.2.15

Multiply $3$ by $81$ .

$0 - (\frac{243}{4} - 3 \cdot 27)$

Step 5.2.2.16

Multiply $- 3$ by $27$ .

$0 - (\frac{243}{4} - 81)$

Step 5.2.2.17

To write $- 81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$0 - (\frac{243}{4} - 81 \cdot \frac{4}{4})$

Step 5.2.2.18

Combine $- 81$ and $\frac{4}{4}$ .

$0 - (\frac{243}{4} + \frac{- 81 \cdot 4}{4})$

Step 5.2.2.19

Combine the numerators over the common denominator.

$0 - \frac{243 - 81 \cdot 4}{4}$

Step 5.2.2.20

Simplify the numerator.

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Step 5.2.2.20.1

Multiply $- 81$ by $4$ .

$0 - \frac{243 - 324}{4}$

Step 5.2.2.20.2

Subtract $324$ from $243$ .

$0 - \frac{- 81}{4}$

Step 5.2.2.21

Move the negative in front of the fraction.

$0 - - \frac{81}{4}$

Step 5.2.2.22

Multiply $- 1$ by $- 1$ .

$0 + 1 (\frac{81}{4})$

Step 5.2.2.23

Multiply $\frac{81}{4}$ by $1$ .

$0 + \frac{81}{4}$

Step 5.2.2.24

Add $0$ and $\frac{81}{4}$ .

$\frac{81}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{81}{4}$

Decimal Form:

$20.25$

Mixed Number Form:

$20 \frac{1}{4}$

Step 7