Calculus Examples

$\int_{1}^{8} (\sqrt[3]{x} - \frac{1}{x}) d x$

Step 1

Remove parentheses.

$\int_{1}^{8} \sqrt[3]{x} - \frac{1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{8} \sqrt[3]{x} d x + \int_{1}^{8} - \frac{1}{x} d x$

Step 3

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

$\int_{1}^{8} x^{\frac{1}{3}} d x + \int_{1}^{8} - \frac{1}{x} d x$

Step 4

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

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{8} + \int_{1}^{8} - \frac{1}{x} d x$

Step 5

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

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{8} - \int_{1}^{8} \frac{1}{x} d x$

Step 6

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{8} - (\ln (| x |)]_{1}^{8})$

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}}$ at $8$ and at $1$ .

$(\frac{3}{4} \cdot 8^{\frac{4}{3}}) - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| x |)]_{1}^{8})$

Step 7.1.2

Evaluate $\ln (| x |)$ at $8$ and at $1$ .

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

Step 7.1.3

Simplify.

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Step 7.1.3.1

Rewrite $8$ as $2^{3}$ .

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

Step 7.1.3.2

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

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

Step 7.1.3.3

Cancel the common factor of $3$ .

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Step 7.1.3.3.1

Cancel the common factor.

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

Step 7.1.3.3.2

Rewrite the expression.

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

Step 7.1.3.4

Raise $2$ to the power of $4$ .

$\frac{3}{4} \cdot 16 - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.5

Combine $\frac{3}{4}$ and $16$ .

$\frac{3 \cdot 16}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.6

Multiply $3$ by $16$ .

$\frac{48}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.7

Cancel the common factor of $48$ and $4$ .

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Step 7.1.3.7.1

Factor $4$ out of $48$ .

$\frac{4 \cdot 12}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.7.2

Cancel the common factors.

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Step 7.1.3.7.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 12}{4 (1)} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.7.2.2

Cancel the common factor.

$\frac{4 \cdot 12}{4 \cdot 1} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.7.2.3

Rewrite the expression.

$\frac{12}{1} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.7.2.4

Divide $12$ by $1$ .

$12 - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.8

One to any power is one.

$12 - \frac{3}{4} \cdot 1 - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.9

Multiply $- 1$ by $1$ .

$12 - \frac{3}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.10

To write $12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$12 \cdot \frac{4}{4} - \frac{3}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.11

Combine $12$ and $\frac{4}{4}$ .

$\frac{12 \cdot 4}{4} - \frac{3}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.12

Combine the numerators over the common denominator.

$\frac{12 \cdot 4 - 3}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.13

Simplify the numerator.

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Step 7.1.3.13.1

Multiply $12$ by $4$ .

$\frac{48 - 3}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.13.2

Subtract $3$ from $48$ .

$\frac{45}{4} - (\ln (| 8 |) - \ln (| 1 |))$

Step 7.1.3.14

To write $- (\ln (| 8 |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{45}{4} - (\ln (| 8 |) - \ln (| 1 |)) \cdot \frac{4}{4}$

Step 7.1.3.15

Combine $- (\ln (| 8 |) - \ln (| 1 |))$ and $\frac{4}{4}$ .

$\frac{45}{4} + \frac{- (\ln (| 8 |) - \ln (| 1 |)) \cdot 4}{4}$

Step 7.1.3.16

Combine the numerators over the common denominator.

$\frac{45 - (\ln (| 8 |) - \ln (| 1 |)) \cdot 4}{4}$

Step 7.1.3.17

Multiply $4$ by $- 1$ .

$\frac{45 - 4 (\ln (| 8 |) - \ln (| 1 |))}{4}$

Step 7.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{45 - 4 \ln (\frac{| 8 |}{| 1 |})}{4}$

Step 7.3

Simplify.

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Step 7.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{45 - 4 \ln (\frac{8}{| 1 |})}{4}$

Step 7.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{45 - 4 \ln (\frac{8}{1})}{4}$

Step 7.3.3

Divide $8$ by $1$ .

$\frac{45 - 4 \ln (8)}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{45 - 4 \ln (8)}{4}$

Decimal Form:

$9.17055845 \dots$

Step 9