Calculus Examples

$\int x (x^{2} + 7) (\frac{1}{3}) d x$

Step 1

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

$\int \frac{x}{3} (x^{2} + 7) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\frac{x}{3}$ .

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

Step 2.1.2

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

$\frac{1}{3} \frac{d}{d x} [x]$

Step 2.1.3

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

$\frac{1}{3} \cdot 1$

Step 2.1.4

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

$\frac{1}{3}$

Step 2.2

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

$\int 3 u_{1} ({(3 u_{1})}^{2} + 7) d u_{1}$

Step 3

Simplify.

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Step 3.1

Factor $3$ out of $3 u_{1}$ .

$\int 3 u_{1} ({(3 (u_{1}))}^{2} + 7) d u_{1}$

Step 3.2

Apply the product rule to $3 (u_{1})$ .

$\int 3 u_{1} (3^{2} {u_{1}}^{2} + 7) d u_{1}$

Step 3.3

Raise $3$ to the power of $2$ .

$\int 3 u_{1} (9 {u_{1}}^{2} + 7) d u_{1}$

Step 4

Let $u_{2} = 9 {u_{1}}^{2} + 7$ . Then $d u_{2} = 18 u_{1} d u_{1}$ , so $\frac{1}{18} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.1

Let $u_{2} = 9 {u_{1}}^{2} + 7$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 4.1.1

Differentiate $9 {u_{1}}^{2} + 7$ .

$\frac{d}{d u_{1}} [9 {u_{1}}^{2} + 7]$

Step 4.1.2

By the Sum Rule, the derivative of $9 {u_{1}}^{2} + 7$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [9 {u_{1}}^{2}] + \frac{d}{d u_{1}} [7]$ .

$\frac{d}{d u_{1}} [9 {u_{1}}^{2}] + \frac{d}{d u_{1}} [7]$

Step 4.1.3

Evaluate $\frac{d}{d u_{1}} [9 {u_{1}}^{2}]$ .

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Step 4.1.3.1

Since $9$ is constant with respect to $u_{1}$ , the derivative of $9 {u_{1}}^{2}$ with respect to $u_{1}$ is $9 \frac{d}{d u_{1}} [{u_{1}}^{2}]$ .

$9 \frac{d}{d u_{1}} [{u_{1}}^{2}] + \frac{d}{d u_{1}} [7]$

Step 4.1.3.2

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

$9 (2 u_{1}) + \frac{d}{d u_{1}} [7]$

Step 4.1.3.3

Multiply $2$ by $9$ .

$18 u_{1} + \frac{d}{d u_{1}} [7]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $7$ is constant with respect to $u_{1}$ , the derivative of $7$ with respect to $u_{1}$ is $0$ .

$18 u_{1} + 0$

Step 4.1.4.2

Add $18 u_{1}$ and $0$ .

$18 u_{1}$

Step 4.2

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

$\int 3 u_{2} \frac{1}{18} d u_{2}$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{18}$ and $3$ .

$\int \frac{3}{18} u_{2} d u_{2}$

Step 5.2

Combine $\frac{3}{18}$ and $u_{2}$ .

$\int \frac{3 u_{2}}{18} d u_{2}$

Step 5.3

Cancel the common factor of $3$ and $18$ .

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Step 5.3.1

Factor $3$ out of $3 u_{2}$ .

$\int \frac{3 (u_{2})}{18} d u_{2}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $3$ out of $18$ .

$\int \frac{3 u_{2}}{3 \cdot 6} d u_{2}$

Step 5.3.2.2

Cancel the common factor.

$\int \frac{3 u_{2}}{3 \cdot 6} d u_{2}$

Step 5.3.2.3

Rewrite the expression.

$\int \frac{u_{2}}{6} d u_{2}$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u_{2}$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int u_{2} d u_{2}$

Step 7

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

$\frac{1}{6} (\frac{1}{2} {u_{2}}^{2} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{1}{6} (\frac{1}{2} {u_{2}}^{2} + C)$ as $\frac{1}{6} \cdot \frac{1}{2} {u_{2}}^{2} + C$ .

$\frac{1}{6} \cdot \frac{1}{2} {u_{2}}^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{2}$ .

$\frac{1}{6 \cdot 2} {u_{2}}^{2} + C$

Step 8.2.2

Multiply $6$ by $2$ .

$\frac{1}{12} {u_{2}}^{2} + C$

Step 9

Substitute back in for each integration substitution variable.

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Step 9.1

Replace all occurrences of $u_{2}$ with $9 {u_{1}}^{2} + 7$ .

$\frac{1}{12} {(9 {u_{1}}^{2} + 7)}^{2} + C$

Step 9.2

Replace all occurrences of $u_{1}$ with $\frac{x}{3}$ .

$\frac{1}{12} {(9 {(\frac{x}{3})}^{2} + 7)}^{2} + C$

Step 10

Reorder terms.

$\frac{1}{12} {(9 {(\frac{1}{3} x)}^{2} + 7)}^{2} + C$