Calculus Examples

$\int (\frac{{(\sqrt{5} - \sqrt{x})}^{2}}{\sqrt{x}}) d x = 10 \sqrt{x} - 2 x \sqrt{5} + \frac{2}{3} \sqrt{x^{3}} + C$

Step 1

Remove parentheses.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Move the negative in front of the fraction.

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

Step 3

Let $u = \sqrt{5} - x^{\frac{1}{2}}$ . Then $d u = - \frac{1}{2 x^{\frac{1}{2}}} d x$ , so $1 d u = - \frac{1}{2 x^{\frac{1}{2}}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \sqrt{5} - x^{\frac{1}{2}}$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $\sqrt{5} - x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\sqrt{5} - x^{\frac{1}{2}}]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $\sqrt{5} - x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{5}] + \frac{d}{d x} [- x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [\sqrt{5}] + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 3.1.2.2

Since $\sqrt{5}$ is constant with respect to $x$ , the derivative of $\sqrt{5}$ with respect to $x$ is $0$ .

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

Step 3.1.3

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

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Step 3.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$0 - \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$0 - (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.1.3.4

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

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

Step 3.1.3.5

Combine the numerators over the common denominator.

$0 - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.1.3.6

Simplify the numerator.

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Step 3.1.3.6.1

Multiply $- 1$ by $2$ .

$0 - (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.1.3.6.2

Subtract $2$ from $1$ .

$0 - (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.1.3.7

Move the negative in front of the fraction.

$0 - (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.1.3.8

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

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

Step 3.1.3.9

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.1.4

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from $0$ .

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

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$\int - 1 \cdot 2 u^{2} d u$

Step 4

Multiply $- 1$ by $2$ .

$\int - 2 u^{2} d u$

Step 5

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$- 2 \int u^{2} d u$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $- 2 (\frac{1}{3} u^{3} + C)$ as $- 2 (\frac{1}{3}) u^{3} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 7.2.2

Move the negative in front of the fraction.

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

Step 8

Replace all occurrences of $u$ with $\sqrt{5} - x^{\frac{1}{2}}$ .

$- \frac{2}{3} {(\sqrt{5} - x^{\frac{1}{2}})}^{3} + C$