Calculus Examples

$\int_{1}^{32} \sqrt[5]{x} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\int_{1}^{32} x^{\frac{1}{5}} d x$

Step 2

By the Power Rule, the integral of $x^{\frac{1}{5}}$ with respect to $x$ is $\frac{5}{6} x^{\frac{6}{5}}$ .

$\frac{5}{6} x^{\frac{6}{5}}]_{1}^{32}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{5}{6} x^{\frac{6}{5}}$ at $32$ and at $1$ .

$(\frac{5}{6} \cdot 32^{\frac{6}{5}}) - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $32$ as $2^{5}$ .

$\frac{5}{6} \cdot {(2^{5})}^{\frac{6}{5}} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.2

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

$\frac{5}{6} \cdot 2^{5 (\frac{6}{5})} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.3

Cancel the common factor of $5$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{5}{6} \cdot 2^{5 (\frac{6}{5})} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.3.2

Rewrite the expression.

$\frac{5}{6} \cdot 2^{6} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.4

Raise $2$ to the power of $6$ .

$\frac{5}{6} \cdot 64 - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.5

Combine $\frac{5}{6}$ and $64$ .

$\frac{5 \cdot 64}{6} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.6

Multiply $5$ by $64$ .

$\frac{320}{6} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.7

Cancel the common factor of $320$ and $6$ .

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Step 3.2.7.1

Factor $2$ out of $320$ .

$\frac{2 (160)}{6} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 160}{2 \cdot 3} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.7.2.2

Cancel the common factor.

$\frac{2 \cdot 160}{2 \cdot 3} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.7.2.3

Rewrite the expression.

$\frac{160}{3} - \frac{5}{6} \cdot 1^{\frac{6}{5}}$

Step 3.2.8

One to any power is one.

$\frac{160}{3} - \frac{5}{6} \cdot 1$

Step 3.2.9

Multiply $- 1$ by $1$ .

$\frac{160}{3} - \frac{5}{6}$

Step 3.2.10

To write $\frac{160}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{160}{3} \cdot \frac{2}{2} - \frac{5}{6}$

Step 3.2.11

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.11.1

Multiply $\frac{160}{3}$ by $\frac{2}{2}$ .

$\frac{160 \cdot 2}{3 \cdot 2} - \frac{5}{6}$

Step 3.2.11.2

Multiply $3$ by $2$ .

$\frac{160 \cdot 2}{6} - \frac{5}{6}$

Step 3.2.12

Combine the numerators over the common denominator.

$\frac{160 \cdot 2 - 5}{6}$

Step 3.2.13

Simplify the numerator.

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Step 3.2.13.1

Multiply $160$ by $2$ .

$\frac{320 - 5}{6}$

Step 3.2.13.2

Subtract $5$ from $320$ .

$\frac{315}{6}$

Step 3.2.14

Cancel the common factor of $315$ and $6$ .

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Step 3.2.14.1

Factor $3$ out of $315$ .

$\frac{3 (105)}{6}$

Step 3.2.14.2

Cancel the common factors.

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Step 3.2.14.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 105}{3 \cdot 2}$

Step 3.2.14.2.2

Cancel the common factor.

$\frac{3 \cdot 105}{3 \cdot 2}$

Step 3.2.14.2.3

Rewrite the expression.

$\frac{105}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{105}{2}$

Decimal Form:

$52.5$

Mixed Number Form:

$52 \frac{1}{2}$

Step 5