Calculus Examples

$\int_{0}^{\sqrt{19}} 9 x \sqrt[3]{8 + x^{2}} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{0}^{\sqrt{19}} x \sqrt[3]{8 + x^{2}} d x$

Step 2

Let $u = 8 + x^{2}$ . 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 2.1

Let $u = 8 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $8 + x^{2}$ .

$\frac{d}{d x} [8 + x^{2}]$

Step 2.1.2

By the Sum Rule, the derivative of $8 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [8] + \frac{d}{d x} [x^{2}]$

Step 2.1.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

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

Step 2.1.4

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 2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.2

Substitute the lower limit in for $x$ in $u = 8 + x^{2}$ .

$u_{lower} = 8 + 0^{2}$

Step 2.3

Simplify.

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Step 2.3.1

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

$u_{lower} = 8 + 0$

Step 2.3.2

Add $8$ and $0$ .

$u_{lower} = 8$

Step 2.4

Substitute the upper limit in for $x$ in $u = 8 + x^{2}$ .

$u_{upper} = 8 + {\sqrt{19}}^{2}$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 2.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$u_{upper} = 8 + {(19^{\frac{1}{2}})}^{2}$

Step 2.5.1.2

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

$u_{upper} = 8 + 19^{\frac{1}{2} \cdot 2}$

Step 2.5.1.3

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

$u_{upper} = 8 + 19^{\frac{2}{2}}$

Step 2.5.1.4

Cancel the common factor of $2$ .

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Step 2.5.1.4.1

Cancel the common factor.

$u_{upper} = 8 + 19^{\frac{2}{2}}$

Step 2.5.1.4.2

Rewrite the expression.

$u_{upper} = 8 + 19^{1}$

Step 2.5.1.5

Evaluate the exponent.

$u_{upper} = 8 + 19$

Step 2.5.2

Add $8$ and $19$ .

$u_{upper} = 27$

Step 2.6

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

$u_{lower} = 8$

$u_{upper} = 27$

Step 2.7

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

$9 \int_{8}^{27} \sqrt[3]{u} \frac{1}{2} d u$

Step 3

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

$9 \int_{8}^{27} \frac{\sqrt[3]{u}}{2} d u$

Step 4

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

$9 (\frac{1}{2} \int_{8}^{27} \sqrt[3]{u} d u)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} \int_{8}^{27} \sqrt[3]{u} d u$

Step 5.2

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

$\frac{9}{2} \int_{8}^{27} u^{\frac{1}{3}} d u$

Step 6

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

$\frac{9}{2} \frac{3}{4} u^{\frac{4}{3}}]_{8}^{27}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{3}{4} u^{\frac{4}{3}}$ at $27$ and at $8$ .

$\frac{9}{2} ((\frac{3}{4} \cdot 27^{\frac{4}{3}}) - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2

Simplify.

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Step 7.2.1

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

$\frac{9}{2} (\frac{3}{4} \cdot {(3^{3})}^{\frac{4}{3}} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.2

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

$\frac{9}{2} (\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.3

Cancel the common factor of $3$ .

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Step 7.2.3.1

Cancel the common factor.

$\frac{9}{2} (\frac{3}{4} \cdot 3^{3 (\frac{4}{3})} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.3.2

Rewrite the expression.

$\frac{9}{2} (\frac{3}{4} \cdot 3^{4} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.4

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

$\frac{9}{2} (\frac{3}{4} \cdot 81 - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.5

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

$\frac{9}{2} (\frac{3 \cdot 81}{4} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.6

Multiply $3$ by $81$ .

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot 8^{\frac{4}{3}})$

Step 7.2.7

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

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot {(2^{3})}^{\frac{4}{3}})$

Step 7.2.8

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

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot 2^{3 (\frac{4}{3})})$

Step 7.2.9

Cancel the common factor of $3$ .

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Step 7.2.9.1

Cancel the common factor.

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot 2^{3 (\frac{4}{3})})$

Step 7.2.9.2

Rewrite the expression.

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot 2^{4})$

Step 7.2.10

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

$\frac{9}{2} (\frac{243}{4} - \frac{3}{4} \cdot 16)$

Step 7.2.11

Multiply $16$ by $- 1$ .

$\frac{9}{2} (\frac{243}{4} - 16 (\frac{3}{4}))$

Step 7.2.12

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

$\frac{9}{2} (\frac{243}{4} + \frac{- 16 \cdot 3}{4})$

Step 7.2.13

Multiply $- 16$ by $3$ .

$\frac{9}{2} (\frac{243}{4} + \frac{- 48}{4})$

Step 7.2.14

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

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Step 7.2.14.1

Factor $4$ out of $- 48$ .

$\frac{9}{2} (\frac{243}{4} + \frac{4 \cdot - 12}{4})$

Step 7.2.14.2

Cancel the common factors.

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Step 7.2.14.2.1

Factor $4$ out of $4$ .

$\frac{9}{2} (\frac{243}{4} + \frac{4 \cdot - 12}{4 (1)})$

Step 7.2.14.2.2

Cancel the common factor.

$\frac{9}{2} (\frac{243}{4} + \frac{4 \cdot - 12}{4 \cdot 1})$

Step 7.2.14.2.3

Rewrite the expression.

$\frac{9}{2} (\frac{243}{4} + \frac{- 12}{1})$

Step 7.2.14.2.4

Divide $- 12$ by $1$ .

$\frac{9}{2} (\frac{243}{4} - 12)$

Step 7.2.15

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{9}{2} (\frac{243}{4} - 12 \cdot \frac{4}{4})$

Step 7.2.16

Combine $- 12$ and $\frac{4}{4}$ .

$\frac{9}{2} (\frac{243}{4} + \frac{- 12 \cdot 4}{4})$

Step 7.2.17

Combine the numerators over the common denominator.

$\frac{9}{2} \cdot \frac{243 - 12 \cdot 4}{4}$

Step 7.2.18

Simplify the numerator.

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Step 7.2.18.1

Multiply $- 12$ by $4$ .

$\frac{9}{2} \cdot \frac{243 - 48}{4}$

Step 7.2.18.2

Subtract $48$ from $243$ .

$\frac{9}{2} \cdot \frac{195}{4}$

Step 7.2.19

Multiply $\frac{9}{2}$ by $\frac{195}{4}$ .

$\frac{9 \cdot 195}{2 \cdot 4}$

Step 7.2.20

Multiply $9$ by $195$ .

$\frac{1755}{2 \cdot 4}$

Step 7.2.21

Multiply $2$ by $4$ .

$\frac{1755}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1755}{8}$

Decimal Form:

$219.375$

Mixed Number Form:

$219 \frac{3}{8}$

Step 9