Calculus Examples

$\int_{1}^{27} (\sqrt[3]{x} - \frac{1}{x}) d x$

Step 1

Remove parentheses.

$\int_{1}^{27} \sqrt[3]{x} - \frac{1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{27} \sqrt[3]{x} d x + \int_{1}^{27} - \frac{1}{x} d x$

Step 3

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

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

Step 4

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

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{27} + \int_{1}^{27} - \frac{1}{x} d x$

Step 5

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

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{27} - \int_{1}^{27} \frac{1}{x} d x$

Step 6

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

$\frac{3}{4} x^{\frac{4}{3}}]_{1}^{27} - (\ln (| x |)]_{1}^{27})$

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 $27$ and at $1$ .

$(\frac{3}{4} \cdot 27^{\frac{4}{3}}) - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| x |)]_{1}^{27})$

Step 7.1.2

Evaluate $\ln (| x |)$ at $27$ and at $1$ .

$\frac{3}{4} \cdot 27^{\frac{4}{3}} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3

Simplify.

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Step 7.1.3.1

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

$\frac{3}{4} \cdot {(3^{3})}^{\frac{4}{3}} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.2

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

$\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

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.

$\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.3.2

Rewrite the expression.

$\frac{3}{4} \cdot 3^{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.4

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

$\frac{3}{4} \cdot 81 - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.5

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

$\frac{3 \cdot 81}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.6

Multiply $3$ by $81$ .

$\frac{243}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.7

One to any power is one.

$\frac{243}{4} - \frac{3}{4} \cdot 1 - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.8

Multiply $- 1$ by $1$ .

$\frac{243}{4} - \frac{3}{4} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.9

Combine the numerators over the common denominator.

$\frac{243 - 3}{4} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.10

Subtract $3$ from $243$ .

$\frac{240}{4} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.11

Cancel the common factor of $240$ and $4$ .

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Step 7.1.3.11.1

Factor $4$ out of $240$ .

$\frac{4 \cdot 60}{4} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.11.2

Cancel the common factors.

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Step 7.1.3.11.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 60}{4 (1)} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.11.2.2

Cancel the common factor.

$\frac{4 \cdot 60}{4 \cdot 1} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.11.2.3

Rewrite the expression.

$\frac{60}{1} - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.1.3.11.2.4

Divide $60$ by $1$ .

$60 - (\ln (| 27 |) - \ln (| 1 |))$

Step 7.2

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

$60 - \ln (\frac{| 27 |}{| 1 |})$

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 $27$ is $27$ .

$60 - \ln (\frac{27}{| 1 |})$

Step 7.3.2

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

$60 - \ln (\frac{27}{1})$

Step 7.3.3

Divide $27$ by $1$ .

$60 - \ln (27)$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (27)$ as $\ln (3^{3})$ .

$60 - \ln (3^{3})$

Step 8.2

Expand $\ln (3^{3})$ by moving $3$ outside the logarithm.

$60 - (3 \ln (3))$

Step 8.3

Multiply $3$ by $- 1$ .

$60 - 3 \ln (3)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$60 - 3 \ln (3)$

Decimal Form:

$56.70416313 \dots$

Step 10