Calculus Examples

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

Step 1

Let $u_{1} = 2 x + 1$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 x + 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 x + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

Let $u_{2} = \frac{u_{1}}{2} - \frac{1}{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 3.1.1

Differentiate $\frac{u_{1}}{2} - \frac{1}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2} - \frac{1}{2}]$

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

Since $- \frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $- \frac{1}{2}$ with respect to $u_{1}$ is $0$ .

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

Step 3.1.4.2

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

$\frac{1}{2}$

Step 3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 4

Let $u_{3} = 2 u_{2} + 1$ . Then $d u_{3} = 2 d u_{2}$ , so $\frac{1}{2} d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 4.1

Let $u_{3} = 2 u_{2} + 1$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 4.1.1

Differentiate $2 u_{2} + 1$ .

$\frac{d}{d u_{2}} [2 u_{2} + 1]$

Step 4.1.2

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

$\frac{d}{d u_{2}} [2 u_{2}] + \frac{d}{d u_{2}} [1]$

Step 4.1.3

Evaluate $\frac{d}{d u_{2}} [2 u_{2}]$ .

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Step 4.1.3.1

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

$2 \frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [1]$

Step 4.1.3.2

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

$2 \cdot 1 + \frac{d}{d u_{2}} [1]$

Step 4.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d u_{2}} [1]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $1$ is constant with respect to $u_{2}$ , the derivative of $1$ with respect to $u_{2}$ is $0$ .

$2 + 0$

Step 4.1.4.2

Add $2$ and $0$ .

$2$

Step 4.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\int ({(\frac{u_{3}}{2} - \frac{1}{2})}^{2} - 1) {u_{3}}^{\frac{1}{2}} \frac{1}{2} d u_{3}$

Step 5

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

$\int ({(\frac{u_{3}}{2} - \frac{1}{2})}^{2} - 1) \frac{{u_{3}}^{\frac{1}{2}}}{2} d u_{3}$

Step 6

Simplify.

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Step 6.1

Rewrite ${(\frac{u_{3}}{2} - \frac{1}{2})}^{2}$ as $(\frac{u_{3}}{2} - \frac{1}{2}) (\frac{u_{3}}{2} - \frac{1}{2})$ .

$\int ((\frac{u_{3}}{2} - \frac{1}{2}) (\frac{u_{3}}{2} - \frac{1}{2}) - 1) \frac{{u_{3}}^{\frac{1}{2}}}{2} d u_{3}$

Step 6.2