Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Substitute and simplify.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor of $3$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

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

Step 3.2.4

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

$\frac{3}{4} \cdot 16 - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.5

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

$\frac{3 \cdot 16}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.6

Multiply $3$ by $16$ .

$\frac{48}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.7

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

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Step 3.2.7.1

Factor $4$ out of $48$ .

$\frac{4 \cdot 12}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

Factor $4$ out of $4$ .

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

Step 3.2.7.2.2

Cancel the common factor.

$\frac{4 \cdot 12}{4 \cdot 1} - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.7.2.3

Rewrite the expression.

$\frac{12}{1} - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.7.2.4

Divide $12$ by $1$ .

$12 - \frac{3}{4} \cdot 1^{\frac{4}{3}}$

Step 3.2.8

One to any power is one.

$12 - \frac{3}{4} \cdot 1$

Step 3.2.9

Multiply $- 1$ by $1$ .

$12 - \frac{3}{4}$

Step 3.2.10

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

$12 \cdot \frac{4}{4} - \frac{3}{4}$

Step 3.2.11

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

$\frac{12 \cdot 4}{4} - \frac{3}{4}$

Step 3.2.12

Combine the numerators over the common denominator.

$\frac{12 \cdot 4 - 3}{4}$

Step 3.2.13

Simplify the numerator.

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Step 3.2.13.1

Multiply $12$ by $4$ .

$\frac{48 - 3}{4}$

Step 3.2.13.2

Subtract $3$ from $48$ .

$\frac{45}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{45}{4}$

Decimal Form:

$11.25$

Mixed Number Form:

$11 \frac{1}{4}$

Step 5