Calculus Examples

$\int_{- 2}^{2} \sqrt[3]{8 x} + \frac{4 x}{x^{3}} d x$

Step 1

Cancel the common factor of $x$ and $x^{3}$ .

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Step 1.1

Factor $x$ out of $4 x$ .

$\int_{- 2}^{2} \sqrt[3]{8 x} + \frac{x \cdot 4}{x^{3}} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $x$ out of $x^{3}$ .

$\int_{- 2}^{2} \sqrt[3]{8 x} + \frac{x \cdot 4}{x \cdot x^{2}} d x$

Step 1.2.2

Cancel the common factor.

$\int_{- 2}^{2} \sqrt[3]{8 x} + \frac{x \cdot 4}{x \cdot x^{2}} d x$

Step 1.2.3

Rewrite the expression.

$\int_{- 2}^{2} \sqrt[3]{8 x} + \frac{4}{x^{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{2} \sqrt[3]{8 x} d x + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 3

Let $u = 8 x$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 3.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

Step 3.1.3

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

$8 \cdot 1$

Step 3.1.4

Multiply $8$ by $1$ .

$8$

Step 3.2

Substitute the lower limit in for $x$ in $u = 8 x$ .

$u_{lower} = 8 \cdot - 2$

Step 3.3

Multiply $8$ by $- 2$ .

$u_{lower} = - 16$

Step 3.4

Substitute the upper limit in for $x$ in $u = 8 x$ .

$u_{upper} = 8 \cdot 2$

Step 3.5

Multiply $8$ by $2$ .

$u_{upper} = 16$

Step 3.6

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

$u_{lower} = - 16$

$u_{upper} = 16$

Step 3.7

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

$\int_{- 16}^{16} \sqrt[3]{u} \frac{1}{8} d u + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 4

Combine $\sqrt[3]{u}$ and $\frac{1}{8}$ .

$\int_{- 16}^{16} \frac{\sqrt[3]{u}}{8} d u + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 5

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int_{- 16}^{16} \sqrt[3]{u} d u + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 6

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

$\frac{1}{8} \int_{- 16}^{16} u^{\frac{1}{3}} d u + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 7

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

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + \int_{- 2}^{2} \frac{4}{x^{2}} d x$

Step 8

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + 4 \int_{- 2}^{2} \frac{1}{x^{2}} d x$

Step 9

Apply basic rules of exponents.

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Step 9.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + 4 \int_{- 2}^{2} {(x^{2})}^{- 1} d x$

Step 9.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 9.2.1

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

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + 4 \int_{- 2}^{2} x^{2 \cdot - 1} d x$

Step 9.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + 4 \int_{- 2}^{2} x^{- 2} d x$

Step 10

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$\frac{1}{8} \frac{3}{4} u^{\frac{4}{3}}]_{- 16}^{16} + 4 (- x^{- 1}]_{- 2}^{2})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{3}{4} u^{\frac{4}{3}}$ at $16$ and at $- 16$ .

$\frac{1}{8} ((\frac{3}{4} \cdot 16^{\frac{4}{3}}) - \frac{3}{4} {(- 16)}^{\frac{4}{3}}) + 4 (- x^{- 1}]_{- 2}^{2})$

Step 11.2

Evaluate $- x^{- 1}$ at $2$ and at $- 2$ .

$\frac{1}{8} ((\frac{3}{4} \cdot 16^{\frac{4}{3}}) - \frac{3}{4} {(- 16)}^{\frac{4}{3}}) + 4 ((- 2^{- 1}) + {(- 2)}^{- 1})$

Step 11.3

Simplify.

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Step 11.3.1

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

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3}{4} {(- 16)}^{\frac{4}{3}}) + 4 ((- 2^{- 1}) + {(- 2)}^{- 1})$

Step 11.3.2

Combine ${(- 16)}^{\frac{4}{3}}$ and $\frac{3}{4}$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{{(- 16)}^{\frac{4}{3}} \cdot 3}{4}) + 4 ((- 2^{- 1}) + {(- 2)}^{- 1})$

Step 11.3.3

Move $3$ to the left of ${(- 16)}^{\frac{4}{3}}$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 \cdot {(- 16)}^{\frac{4}{3}}}{4}) + 4 ((- 2^{- 1}) + {(- 2)}^{- 1})$

Step 11.3.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (- \frac{1}{2} + {(- 2)}^{- 1})$

Step 11.3.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (- \frac{1}{2} + \frac{1}{- 2})$

Step 11.3.6

Move the negative in front of the fraction.

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (- \frac{1}{2} - \frac{1}{2})$

Step 11.3.7

Subtract $\frac{1}{2}$ from $- \frac{1}{2}$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (- 2 (\frac{1}{2}))$

Step 11.3.8

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

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (\frac{- 2}{2})$

Step 11.3.9

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

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Step 11.3.9.1

Factor $2$ out of $- 2$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 \frac{2 \cdot - 1}{2}$

Step 11.3.9.2

Cancel the common factors.

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Step 11.3.9.2.1

Factor $2$ out of $2$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 \frac{2 \cdot - 1}{2 (1)}$

Step 11.3.9.2.2

Cancel the common factor.

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 \frac{2 \cdot - 1}{2 \cdot 1}$

Step 11.3.9.2.3

Rewrite the expression.

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 (\frac{- 1}{1})$

Step 11.3.9.2.4

Divide $- 1$ by $1$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) + 4 \cdot - 1$

Step 11.3.10

Multiply $4$ by $- 1$ .

$\frac{1}{8} (\frac{3 \cdot 16^{\frac{4}{3}}}{4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) - 4$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$\frac{1}{8} \cdot \frac{3 \cdot 16^{\frac{4}{3}}}{4} + \frac{1}{8} (- \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) - 4$

Step 12.2

Combine.

$\frac{1 (3 \cdot 16^{\frac{4}{3}})}{8 \cdot 4} + \frac{1}{8} (- \frac{3 {(- 16)}^{\frac{4}{3}}}{4}) - 4$

Step 12.3

Multiply $\frac{1}{8} (- \frac{3 {(- 16)}^{\frac{4}{3}}}{4})$ .

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Step 12.3.1

Multiply $\frac{1}{8}$ by $\frac{3 {(- 16)}^{\frac{4}{3}}}{4}$ .

$\frac{1 (3 \cdot 16^{\frac{4}{3}})}{8 \cdot 4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{8 \cdot 4} - 4$

Step 12.3.2

Multiply $8$ by $4$ .

$\frac{1 (3 \cdot 16^{\frac{4}{3}})}{8 \cdot 4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{32} - 4$

Step 12.4

Simplify each term.

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Step 12.4.1

Multiply $3$ by $1$ .

$\frac{3 \cdot 16^{\frac{4}{3}}}{8 \cdot 4} - \frac{3 {(- 16)}^{\frac{4}{3}}}{32} - 4$

Step 12.4.2

Multiply $8$ by $4$ .

$\frac{3 \cdot 16^{\frac{4}{3}}}{32} - \frac{3 {(- 16)}^{\frac{4}{3}}}{32} - 4$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \cdot 16^{\frac{4}{3}}}{32} - \frac{3 {(- 16)}^{\frac{4}{3}}}{32} - 4$

Decimal Form:

$- 4$

Step 14