Calculus Examples

$\int_{1}^{9} \frac{1}{\sqrt[3]{x - 9}} d x$

Step 1

Let $u = x - 9$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 9$ .

$\frac{d}{d x} [x - 9]$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$u_{lower} = 1 - 9$

Step 1.3

Subtract $9$ from $1$ .

$u_{lower} = - 8$

Step 1.4

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

$u_{upper} = 9 - 9$

Step 1.5

Subtract $9$ from $9$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = - 8$

$u_{upper} = 0$

Step 1.7

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

$\int_{- 8}^{0} \frac{1}{\sqrt[3]{u}} d u$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\int_{- 8}^{0} \frac{1}{u^{\frac{1}{3}}} d u$

Step 2.2

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{- 8}^{0} {(u^{\frac{1}{3}})}^{- 1} d u$

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 2.3.1

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

$\int_{- 8}^{0} u^{\frac{1}{3} \cdot - 1} d u$

Step 2.3.2

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

$\int_{- 8}^{0} u^{\frac{- 1}{3}} d u$

Step 2.3.3

Move the negative in front of the fraction.

$\int_{- 8}^{0} u^{- \frac{1}{3}} d u$

Step 3

By the Power Rule, the integral of $u^{- \frac{1}{3}}$ with respect to $u$ is $\frac{3}{2} u^{\frac{2}{3}}$ .

$\frac{3}{2} u^{\frac{2}{3}}]_{- 8}^{0}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{3}{2} u^{\frac{2}{3}}$ at $0$ and at $- 8$ .

$(\frac{3}{2} \cdot 0^{\frac{2}{3}}) - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{3}{2} \cdot {(0^{3})}^{\frac{2}{3}} - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.2

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

$\frac{3}{2} \cdot 0^{3 (\frac{2}{3})} - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.3

Cancel the common factor of $3$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{3}{2} \cdot 0^{3 (\frac{2}{3})} - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.3.2

Rewrite the expression.

$\frac{3}{2} \cdot 0^{2} - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.4

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

$\frac{3}{2} \cdot 0 - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.5

Multiply $\frac{3}{2}$ by $0$ .

$0 - \frac{3}{2} {(- 8)}^{\frac{2}{3}}$

Step 4.2.6

Rewrite $- 8$ as ${(- 2)}^{3}$ .

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

Step 4.2.7

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

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

Step 4.2.8

Cancel the common factor of $3$ .

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Step 4.2.8.1

Cancel the common factor.

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

Step 4.2.8.2

Rewrite the expression.

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

Step 4.2.9

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

$0 - \frac{3}{2} \cdot 4$

Step 4.2.10

Multiply $4$ by $- 1$ .

$0 - 4 (\frac{3}{2})$

Step 4.2.11

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

$0 + \frac{- 4 \cdot 3}{2}$

Step 4.2.12

Multiply $- 4$ by $3$ .

$0 + \frac{- 12}{2}$

Step 4.2.13

Cancel the common factor of $- 12$ and $2$ .

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Step 4.2.13.1

Factor $2$ out of $- 12$ .

$0 + \frac{2 \cdot - 6}{2}$

Step 4.2.13.2

Cancel the common factors.

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Step 4.2.13.2.1

Factor $2$ out of $2$ .

$0 + \frac{2 \cdot - 6}{2 (1)}$

Step 4.2.13.2.2

Cancel the common factor.

$0 + \frac{2 \cdot - 6}{2 \cdot 1}$

Step 4.2.13.2.3

Rewrite the expression.

$0 + \frac{- 6}{1}$

Step 4.2.13.2.4

Divide $- 6$ by $1$ .

$0 - 6$

Step 4.2.14

Subtract $6$ from $0$ .

$- 6$

Step 5