Calculus Examples

\int \frac{{(\sqrt{x} + 2)}^{2}}{5 \sqrt{x}} d x

$\int \frac{{(\sqrt{x} + 2)}^{2}}{5 \sqrt{x}} d x$

Step 1

Since

\frac{1}{5}

$\frac{1}{5}$ is constant with respect to

x

$x$ , move

\frac{1}{5}

$\frac{1}{5}$ out of the integral.

\frac{1}{5} \int \frac{{(\sqrt{x} + 2)}^{2}}{\sqrt{x}} d x

$\frac{1}{5} \int \frac{{(\sqrt{x} + 2)}^{2}}{\sqrt{x}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ .

\frac{1}{5} \int \frac{{(x^{\frac{1}{2}} + 2)}^{2}}{\sqrt{x}} d x

$\frac{1}{5} \int \frac{{(x^{\frac{1}{2}} + 2)}^{2}}{\sqrt{x}} d x$

Step 2.2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ .

\frac{1}{5} \int \frac{{(x^{\frac{1}{2}} + 2)}^{2}}{x^{\frac{1}{2}}} d x

$\frac{1}{5} \int \frac{{(x^{\frac{1}{2}} + 2)}^{2}}{x^{\frac{1}{2}}} d x$

Step 2.3

Move

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ out of the denominator by raising it to the

- 1

$- 1$ power.

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} {(x^{\frac{1}{2}})}^{- 1} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.4

Multiply the exponents in

{(x^{\frac{1}{2}})}^{- 1}

${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 2.4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{\frac{1}{2} \cdot - 1} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{\frac{1}{2} \cdot - 1} d x$

Step 2.4.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

- 1

$- 1$ .

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{\frac{- 1}{2}} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{\frac{- 1}{2}} d x$

Step 2.4.3

Move the negative in front of the fraction.

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x$

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x$

\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x

$\frac{1}{5} \int {(x^{\frac{1}{2}} + 2)}^{2} x^{- \frac{1}{2}} d x$

Step 3

Let

u = x^{\frac{1}{2}} + 2

$u = x^{\frac{1}{2}} + 2$ . Then

d u = \frac{1}{2 x^{\frac{1}{2}}} d x

$d u = \frac{1}{2 x^{\frac{1}{2}}} d x$ , so

- d u = \frac{- 1}{2 x^{\frac{1}{2}}} d x

$- d u = \frac{- 1}{2 x^{\frac{1}{2}}} d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1

Let

u = x^{\frac{1}{2}} + 2

$u = x^{\frac{1}{2}} + 2$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate

x^{\frac{1}{2}} + 2

$x^{\frac{1}{2}} + 2$ .

\frac{d}{d x} [x^{\frac{1}{2}} + 2]

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

Step 3.1.2

By the Sum Rule, the derivative of

x^{\frac{1}{2}} + 2

$x^{\frac{1}{2}} + 2$ with respect to

x

$x$ is

\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [2]

$\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [2]$ .

\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [2]

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

Step 3.1.3

Evaluate

\frac{d}{d x} [x^{\frac{1}{2}}]

$\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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Step 3.1.3.1

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

$n = \frac{1}{2}$ .

\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [2]

$\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [2]$

Step 3.1.3.2

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [2]

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

Step 3.1.3.3

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [2]

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

Step 3.1.3.4

Combine the numerators over the common denominator.

\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [2]

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

Step 3.1.3.5

Simplify the numerator.

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Step 3.1.3.5.1

Multiply

- 1

$- 1$ by

2

$2$ .

\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [2]

$\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [2]$

Step 3.1.3.5.2

Subtract

2

$2$ from

1

$1$ .

\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [2]

$\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [2]$

\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [2]

$\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [2]$

Step 3.1.3.6

Move the negative in front of the fraction.

\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [2]

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [2]$

\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [2]

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [2]$

Step 3.1.4

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2

$2$ with respect to

x

$x$ is

0

$0$ .

\frac{1}{2} x^{- \frac{1}{2}} + 0

$\frac{1}{2} x^{- \frac{1}{2}} + 0$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0$

Step 3.1.5.2

Combine terms.

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Step 3.1.5.2.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{1}{x^{\frac{1}{2}}}

$\frac{1}{x^{\frac{1}{2}}}$ .

\frac{1}{2 x^{\frac{1}{2}}} + 0

$\frac{1}{2 x^{\frac{1}{2}}} + 0$

Step 3.1.5.2.2

Add

\frac{1}{2 x^{\frac{1}{2}}}

$\frac{1}{2 x^{\frac{1}{2}}}$ and

0

$0$ .

\frac{1}{2 x^{\frac{1}{2}}}

$\frac{1}{2 x^{\frac{1}{2}}}$

\frac{1}{2 x^{\frac{1}{2}}}

$\frac{1}{2 x^{\frac{1}{2}}}$

\frac{1}{2 x^{\frac{1}{2}}}

$\frac{1}{2 x^{\frac{1}{2}}}$

\frac{1}{2 x^{\frac{1}{2}}}

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 3.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\frac{1}{5} \int 2 u^{2} d u

$\frac{1}{5} \int 2 u^{2} d u$

\frac{1}{5} \int 2 u^{2} d u

$\frac{1}{5} \int 2 u^{2} d u$

Step 4

Since

2

$2$ is constant with respect to

u

$u$ , move

2

$2$ out of the integral.

\frac{1}{5} (2 \int u^{2} d u)

$\frac{1}{5} (2 \int u^{2} d u)$

Step 5

Combine

2

$2$ and

\frac{1}{5}

$\frac{1}{5}$ .

\frac{2}{5} \int u^{2} d u

$\frac{2}{5} \int u^{2} d u$

Step 6

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

$\frac{1}{3} u^{3}$ .

\frac{2}{5} (\frac{1}{3} u^{3} + C)

$\frac{2}{5} (\frac{1}{3} u^{3} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite

\frac{2}{5} (\frac{1}{3} u^{3} + C)

$\frac{2}{5} (\frac{1}{3} u^{3} + C)$ as

\frac{2}{5} \cdot \frac{1}{3} u^{3} + C

$\frac{2}{5} \cdot \frac{1}{3} u^{3} + C$ .

\frac{2}{5} \cdot \frac{1}{3} u^{3} + C

$\frac{2}{5} \cdot \frac{1}{3} u^{3} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply

\frac{2}{5}

$\frac{2}{5}$ by

\frac{1}{3}

$\frac{1}{3}$ .

\frac{2}{5 \cdot 3} u^{3} + C

$\frac{2}{5 \cdot 3} u^{3} + C$

Step 7.2.2

Multiply

5

$5$ by

3

$3$ .

\frac{2}{15} u^{3} + C

$\frac{2}{15} u^{3} + C$

\frac{2}{15} u^{3} + C

$\frac{2}{15} u^{3} + C$

\frac{2}{15} u^{3} + C

$\frac{2}{15} u^{3} + C$

Step 8

Replace all occurrences of

u

$u$ with

x^{\frac{1}{2}} + 2

$x^{\frac{1}{2}} + 2$ .

\frac{2}{15} {(x^{\frac{1}{2}} + 2)}^{3} + C

$\frac{2}{15} {(x^{\frac{1}{2}} + 2)}^{3} + C$