Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Combine the numerators over the common denominator.

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

Step 2.1.3.5

Simplify the numerator.

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Step 2.1.3.5.1

Multiply $- 1$ by $2$ .

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

Step 2.1.3.5.2

Subtract $2$ from $1$ .

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

Step 2.1.3.6

Move the negative in front of the fraction.

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

Step 2.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 2.1.5

Simplify.

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Step 2.1.5.1

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

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

Step 2.1.5.2

Combine terms.

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Step 2.1.5.2.1

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

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

Step 2.1.5.2.2

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

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 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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