Calculus Examples

$\int x^{2} \sqrt{3 x + 2} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $3 x + 2$ .

$\frac{d}{d x} [3 x + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

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 $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Combine $\frac{{(\frac{u_{1}}{3} - \frac{2}{3})}^{2}}{3}$ and $\sqrt{u_{1}}$ .

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $\frac{u_{1}}{3} - \frac{2}{3}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{3} - \frac{2}{3}]$

Step 5.1.2

By the Sum Rule, the derivative of $\frac{u_{1}}{3} - \frac{2}{3}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [\frac{u_{1}}{3}] + \frac{d}{d u_{1}} [- \frac{2}{3}]$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{3}] + \frac{d}{d u_{1}} [- \frac{2}{3}]$

Step 5.1.3

Evaluate $\frac{d}{d u_{1}} [\frac{u_{1}}{3}]$ .

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Step 5.1.3.1

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{3}$ with respect to $u_{1}$ is $\frac{1}{3} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{3} \frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- \frac{2}{3}]$

Step 5.1.3.2

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

$\frac{1}{3} \cdot 1 + \frac{d}{d u_{1}} [- \frac{2}{3}]$

Step 5.1.3.3

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} + \frac{d}{d u_{1}} [- \frac{2}{3}]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $- \frac{2}{3}$ is constant with respect to $u_{1}$ , the derivative of $- \frac{2}{3}$ with respect to $u_{1}$ is $0$ .

$\frac{1}{3} + 0$

Step 5.1.4.2

Add $\frac{1}{3}$ and $0$ .

$\frac{1}{3}$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

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

Step 6.2

Multiply $3$ by $1$ .

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

Step 6.3

Move $3$ to the left of ${u_{2}}^{2} {(3 u_{2} + 2)}^{\frac{1}{2}}$ .

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

Step 7

Since $3$ is constant with respect to $u_{2}$ , move $3$ out of the integral.

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Cancel the common factor of $3$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

Multiply $\int {u_{2}}^{2} {(3 u_{2} + 2)}^{\frac{1}{2}} d u_{2}$ by $1$ .

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $3 u_{2} + 2$ .

$\frac{d}{d u_{2}} [3 u_{2} + 2]$

Step 9.1.2

By the Sum Rule, the derivative of $3 u_{2} + 2$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [3 u_{2}] + \frac{d}{d u_{2}} [2]$ .

$\frac{d}{d u_{2}} [3 u_{2}] + \frac{d}{d u_{2}} [2]$

Step 9.1.3

Evaluate $\frac{d}{d u_{2}} [3 u_{2}]$ .

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Step 9.1.3.1

Since $3$ is constant with respect to $u_{2}$ , the derivative of $3 u_{2}$ with respect to $u_{2}$ is $3 \frac{d}{d u_{2}} [u_{2}]$ .

$3 \frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [2]$

Step 9.1.3.2

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

$3 \cdot 1 + \frac{d}{d u_{2}} [2]$

Step 9.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d u_{2}} [2]$

Step 9.1.4

Differentiate using the Constant Rule.

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Step 9.1.4.1

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

$3 + 0$

Step 9.1.4.2

Add $3$ and $0$ .

$3$

Step 9.2

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Since $\frac{1}{3}$ is constant with respect to $u_{3}$ , move $\frac{1}{3}$ out of the integral.

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2