Calculus Examples

$\int {(9 x + 2)}^{2} d x$

Step 1

Let $u = 9 x + 2$ . Then $d u = 9 d x$ , so $\frac{1}{9} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 9 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $9 x + 2$ .

$\frac{d}{d x} [9 x + 2]$

Step 1.1.2

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

$\frac{d}{d x} [9 x] + \frac{d}{d x} [2]$

Step 1.1.3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.1.3.1

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

$9 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

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 $n = 1$ .

$9 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.3.3

Multiply $9$ by $1$ .

$9 + \frac{d}{d x} [2]$

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 $0$ .

$9 + 0$

Step 1.1.4.2

Add $9$ and $0$ .

$9$

Step 1.2

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

$\int u^{2} \frac{1}{9} d u$

Step 2

Combine $u^{2}$ and $\frac{1}{9}$ .

$\int \frac{u^{2}}{9} d u$

Step 3

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{9} \int u^{2} d u$

Step 4

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

$\frac{1}{9} (\frac{1}{3} u^{3} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{9} (\frac{1}{3} u^{3} + C)$ as $\frac{1}{9} \cdot \frac{1}{3} u^{3} + C$ .

$\frac{1}{9} \cdot \frac{1}{3} u^{3} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{9}$ by $\frac{1}{3}$ .

$\frac{1}{9 \cdot 3} u^{3} + C$

Step 5.2.2

Multiply $9$ by $3$ .

$\frac{1}{27} u^{3} + C$

Step 6

Replace all occurrences of $u$ with $9 x + 2$ .

$\frac{1}{27} {(9 x + 2)}^{3} + C$