Calculus Examples

$\int_{- 5}^{9} \frac{2 {(2 x + 9)}^{\frac{1}{3}}}{3} d x$

Step 1

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$\frac{2}{3} \int_{- 5}^{9} {(2 x + 9)}^{\frac{1}{3}} d x$

Step 2

Let $u = 2 x + 9$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 x + 9$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x + 9$ .

$\frac{d}{d x} [2 x + 9]$

Step 2.1.2

By the Sum Rule, the derivative of $2 x + 9$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [9]$

Step 2.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

Step 2.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [9]$

Step 2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [9]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 2 x + 9$ .

$u_{lower} = 2 \cdot - 5 + 9$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $2$ by $- 5$ .

$u_{lower} = - 10 + 9$

Step 2.3.2

Add $- 10$ and $9$ .

$u_{lower} = - 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = 2 x + 9$ .

$u_{upper} = 2 \cdot 9 + 9$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $2$ by $9$ .

$u_{upper} = 18 + 9$

Step 2.5.2

Add $18$ and $9$ .

$u_{upper} = 27$

Step 2.6

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

$u_{lower} = - 1$

$u_{upper} = 27$

Step 2.7

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

$\frac{2}{3} \int_{- 1}^{27} u^{\frac{1}{3}} \frac{1}{2} d u$

Step 3

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

$\frac{2}{3} \int_{- 1}^{27} \frac{u^{\frac{1}{3}}}{2} d u$

Step 4

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

$\frac{2}{3} (\frac{1}{2} \int_{- 1}^{27} u^{\frac{1}{3}} d u)$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$\frac{2}{2 \cdot 3} \int_{- 1}^{27} u^{\frac{1}{3}} d u$

Step 5.2

Multiply $2$ by $3$ .

$\frac{2}{6} \int_{- 1}^{27} u^{\frac{1}{3}} d u$

Step 5.3

Cancel the common factor of $2$ and $6$ .

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Step 5.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} \int_{- 1}^{27} u^{\frac{1}{3}} d u$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} \int_{- 1}^{27} u^{\frac{1}{3}} d u$

Step 5.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} \int_{- 1}^{27} u^{\frac{1}{3}} d u$

Step 5.3.2.3

Rewrite the expression.

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

Step 6

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

$\frac{1}{3} \frac{3}{4} u^{\frac{4}{3}}]_{- 1}^{27}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{3}{4} u^{\frac{4}{3}}$ at $27$ and at $- 1$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor of $3$ .

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Step 7.2.3.1

Cancel the common factor.

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

Step 7.2.3.2

Rewrite the expression.

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

Step 7.2.4

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

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

Step 7.2.5

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

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

Step 7.2.6

Multiply $3$ by $81$ .

$\frac{1}{3} (\frac{243}{4} - \frac{3}{4} {(- 1)}^{\frac{4}{3}})$

Step 7.2.7

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\frac{1}{3} (\frac{243}{4} - \frac{3}{4} {({(- 1)}^{3})}^{\frac{4}{3}})$

Step 7.2.8

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

$\frac{1}{3} (\frac{243}{4} - \frac{3}{4} {(- 1)}^{3 (\frac{4}{3})})$

Step 7.2.9

Cancel the common factor of $3$ .

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Step 7.2.9.1

Cancel the common factor.

$\frac{1}{3} (\frac{243}{4} - \frac{3}{4} {(- 1)}^{3 (\frac{4}{3})})$

Step 7.2.9.2

Rewrite the expression.

$\frac{1}{3} (\frac{243}{4} - \frac{3}{4} {(- 1)}^{4})$

Step 7.2.10

Raise $- 1$ to the power of $4$ .

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

Step 7.2.11

Multiply $- 1$ by $1$ .

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

Step 7.2.12

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{243 - 3}{4}$

Step 7.2.13

Subtract $3$ from $243$ .

$\frac{1}{3} \cdot \frac{240}{4}$

Step 7.2.14

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

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Step 7.2.14.1

Factor $4$ out of $240$ .

$\frac{1}{3} \cdot \frac{4 \cdot 60}{4}$

Step 7.2.14.2

Cancel the common factors.

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Step 7.2.14.2.1

Factor $4$ out of $4$ .

$\frac{1}{3} \cdot \frac{4 \cdot 60}{4 (1)}$

Step 7.2.14.2.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{4 \cdot 60}{4 \cdot 1}$

Step 7.2.14.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{60}{1}$

Step 7.2.14.2.4

Divide $60$ by $1$ .

$\frac{1}{3} \cdot 60$

Step 7.2.15

Combine $\frac{1}{3}$ and $60$ .

$\frac{60}{3}$

Step 7.2.16

Cancel the common factor of $60$ and $3$ .

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Step 7.2.16.1

Factor $3$ out of $60$ .

$\frac{3 \cdot 20}{3}$

Step 7.2.16.2

Cancel the common factors.

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Step 7.2.16.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 20}{3 (1)}$

Step 7.2.16.2.2

Cancel the common factor.

$\frac{3 \cdot 20}{3 \cdot 1}$

Step 7.2.16.2.3

Rewrite the expression.

$\frac{20}{1}$

Step 7.2.16.2.4

Divide $20$ by $1$ .

$20$

Step 8