Calculus Examples

$p \int_{0}^{8} {(\sqrt[3]{x})}^{2} d x$

Step 1

Rewrite ${(\sqrt[3]{x})}^{2}$ as $\sqrt[3]{x^{2}}$ .

$p \int_{0}^{8} \sqrt[3]{x^{2}} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$p \int_{0}^{8} x^{\frac{2}{3}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$p \frac{3}{5} x^{\frac{5}{3}}]_{0}^{8}$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$p \frac{3 x^{\frac{5}{3}}}{5}]_{0}^{8}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{3 x^{\frac{5}{3}}}{5}$ at $8$ and at $0$ .

$p ((\frac{3 \cdot 8^{\frac{5}{3}}}{5}) - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$p (\frac{3 \cdot {(2^{3})}^{\frac{5}{3}}}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.2

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

$p (\frac{3 \cdot 2^{3 (\frac{5}{3})}}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.3

Cancel the common factor of $3$ .

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Step 4.2.2.3.1

Cancel the common factor.

$p (\frac{3 \cdot 2^{3 (\frac{5}{3})}}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.3.2

Rewrite the expression.

$p (\frac{3 \cdot 2^{5}}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.4

Raise $2$ to the power of $5$ .

$p (\frac{3 \cdot 32}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.5

Multiply $3$ by $32$ .

$p (\frac{96}{5} - \frac{3 \cdot 0^{\frac{5}{3}}}{5})$

Step 4.2.2.6

Rewrite $0$ as $0^{3}$ .

$p (\frac{96}{5} - \frac{3 \cdot {(0^{3})}^{\frac{5}{3}}}{5})$

Step 4.2.2.7

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

$p (\frac{96}{5} - \frac{3 \cdot 0^{3 (\frac{5}{3})}}{5})$

Step 4.2.2.8

Cancel the common factor of $3$ .

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Step 4.2.2.8.1

Cancel the common factor.

$p (\frac{96}{5} - \frac{3 \cdot 0^{3 (\frac{5}{3})}}{5})$

Step 4.2.2.8.2

Rewrite the expression.

$p (\frac{96}{5} - \frac{3 \cdot 0^{5}}{5})$

Step 4.2.2.9

Raising $0$ to any positive power yields $0$ .

$p (\frac{96}{5} - \frac{3 \cdot 0}{5})$

Step 4.2.2.10

Multiply $3$ by $0$ .

$p (\frac{96}{5} - \frac{0}{5})$

Step 4.2.2.11

Cancel the common factor of $0$ and $5$ .

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Step 4.2.2.11.1

Factor $5$ out of $0$ .

$p (\frac{96}{5} - \frac{5 (0)}{5})$

Step 4.2.2.11.2

Cancel the common factors.

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Step 4.2.2.11.2.1

Factor $5$ out of $5$ .

$p (\frac{96}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 4.2.2.11.2.2

Cancel the common factor.

$p (\frac{96}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 4.2.2.11.2.3

Rewrite the expression.

$p (\frac{96}{5} - \frac{0}{1})$

Step 4.2.2.11.2.4

Divide $0$ by $1$ .

$p (\frac{96}{5} - 0)$

Step 4.2.2.12

Multiply $- 1$ by $0$ .

$p (\frac{96}{5} + 0)$

Step 4.2.2.13

Add $\frac{96}{5}$ and $0$ .

$p \frac{96}{5}$

Step 4.2.2.14

Combine $p$ and $\frac{96}{5}$ .

$\frac{p \cdot 96}{5}$

Step 4.2.2.15

Move $96$ to the left of $p$ .

$\frac{96 p}{5}$

Step 4.3

Reorder terms.

$\frac{96}{5} p$

Step 5

Combine $\frac{96}{5}$ and $p$ .

$\frac{96 p}{5}$

Step 6