Calculus Examples

$\int_{0}^{2} x \sqrt[3]{4 + x^{2}} d x$

Step 1

Let $u = 4 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 + x^{2}$ .

$\frac{d}{d x} [4 + x^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $4 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [x^{2}]$

Step 1.1.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 + 2 x$

Step 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 + x^{2}$ .

$u_{lower} = 4 + 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 4 + 0$

Step 1.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 + x^{2}$ .

$u_{upper} = 4 + 2^{2}$

Step 1.5

Simplify.

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Step 1.5.1

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

$u_{upper} = 4 + 4$

Step 1.5.2

Add $4$ and $4$ .

$u_{upper} = 8$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 8$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4}^{8} \sqrt[3]{u} \frac{1}{2} d u$

Step 2

Combine $\sqrt[3]{u}$ and $\frac{1}{2}$ .

$\int_{4}^{8} \frac{\sqrt[3]{u}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{4}^{8} \sqrt[3]{u} d u$

Step 4

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

$\frac{1}{2} \int_{4}^{8} u^{\frac{1}{3}} d u$

Step 5

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

$\frac{1}{2} \frac{3}{4} u^{\frac{4}{3}}]_{4}^{8}$

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor of $3$ .

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Step 6.2.3.1

Cancel the common factor.

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

Step 6.2.3.2

Rewrite the expression.

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

Step 6.2.4

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

$\frac{1}{2} (\frac{3}{4} \cdot 16 - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.5

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

$\frac{1}{2} (\frac{3 \cdot 16}{4} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.6

Multiply $3$ by $16$ .

$\frac{1}{2} (\frac{48}{4} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.7

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

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Step 6.2.7.1

Factor $4$ out of $48$ .

$\frac{1}{2} (\frac{4 \cdot 12}{4} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.7.2

Cancel the common factors.

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Step 6.2.7.2.1

Factor $4$ out of $4$ .

$\frac{1}{2} (\frac{4 \cdot 12}{4 (1)} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.7.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{4 \cdot 12}{4 \cdot 1} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.7.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{12}{1} - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.7.2.4

Divide $12$ by $1$ .

$\frac{1}{2} (12 - \frac{3}{4} \cdot 4^{\frac{4}{3}})$

Step 6.2.8

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

$\frac{1}{2} (12 - \frac{4^{\frac{4}{3}} \cdot 3}{4})$

Step 6.2.9

Move $3$ to the left of $4^{\frac{4}{3}}$ .

$\frac{1}{2} (12 - \frac{3 \cdot 4^{\frac{4}{3}}}{4})$

Step 6.2.10

Move $4^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{1}{2} (12 - (3 \cdot 4^{\frac{4}{3}} \cdot 4^{- 1}))$

Step 6.2.11

Multiply $4^{\frac{4}{3}}$ by $4^{- 1}$ by adding the exponents.

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Step 6.2.11.1

Move $4^{- 1}$ .

$\frac{1}{2} (12 - (3 \cdot (4^{- 1} \cdot 4^{\frac{4}{3}})))$

Step 6.2.11.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} (12 - (3 \cdot 4^{- 1 + \frac{4}{3}}))$

Step 6.2.11.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{2} (12 - (3 \cdot 4^{- 1 \cdot \frac{3}{3} + \frac{4}{3}}))$

Step 6.2.11.4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{2} (12 - (3 \cdot 4^{\frac{- 1 \cdot 3}{3} + \frac{4}{3}}))$

Step 6.2.11.5

Combine the numerators over the common denominator.

$\frac{1}{2} (12 - (3 \cdot 4^{\frac{- 1 \cdot 3 + 4}{3}}))$

Step 6.2.11.6

Simplify the numerator.

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Step 6.2.11.6.1

Multiply $- 1$ by $3$ .

$\frac{1}{2} (12 - (3 \cdot 4^{\frac{- 3 + 4}{3}}))$

Step 6.2.11.6.2

Add $- 3$ and $4$ .

$\frac{1}{2} (12 - (3 \cdot 4^{\frac{1}{3}}))$

Step 6.2.12

Multiply $3$ by $- 1$ .

$\frac{1}{2} (12 - 3 \cdot 4^{\frac{1}{3}})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} \cdot 12 + \frac{1}{2} (- 3 \cdot 4^{\frac{1}{3}})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $12$ .

$\frac{1}{2} \cdot (2 (6)) + \frac{1}{2} (- 3 \cdot 4^{\frac{1}{3}})$

Step 7.2.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 6) + \frac{1}{2} (- 3 \cdot 4^{\frac{1}{3}})$

Step 7.2.3

Rewrite the expression.

$6 + \frac{1}{2} (- 3 \cdot 4^{\frac{1}{3}})$

Step 7.3

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

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Step 7.3.1

Combine $- 3$ and $\frac{1}{2}$ .

$6 + \frac{- 3}{2} \cdot 4^{\frac{1}{3}}$

Step 7.3.2

Combine $\frac{- 3}{2}$ and $4^{\frac{1}{3}}$ .

$6 + \frac{- 3 \cdot 4^{\frac{1}{3}}}{2}$

Step 7.4

Move the negative in front of the fraction.

$6 - \frac{3 \cdot 4^{\frac{1}{3}}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 - \frac{3 \cdot 4^{\frac{1}{3}}}{2}$

Decimal Form:

$3.61889842 \dots$

Step 9