Calculus Examples

\int {(9 x + 4)}^{2} d x

$\int {(9 x + 4)}^{2} d x$

Step 1

Let

u = 9 x + 4

$u = 9 x + 4$ . Then

d u = 9 d x

$d u = 9 d x$ , so

\frac{1}{9} d u = d x

$\frac{1}{9} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = 9 x + 4

$u = 9 x + 4$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

9 x + 4

$9 x + 4$ .

\frac{d}{d x} [9 x + 4]

$\frac{d}{d x} [9 x + 4]$

Step 1.1.2

By the Sum Rule, the derivative of

9 x + 4

$9 x + 4$ with respect to

x

$x$ is

\frac{d}{d x} [9 x] + \frac{d}{d x} [4]

$\frac{d}{d x} [9 x] + \frac{d}{d x} [4]$ .

\frac{d}{d x} [9 x] + \frac{d}{d x} [4]

$\frac{d}{d x} [9 x] + \frac{d}{d x} [4]$

Step 1.1.3

Evaluate

\frac{d}{d x} [9 x]

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

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Step 1.1.3.1

Since

9

$9$ is constant with respect to

x

$x$ , the derivative of

9 x

$9 x$ with respect to

x

$x$ is

9 \frac{d}{d x} [x]

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

$9 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

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

$n = 1$ .

$9 \cdot 1 + \frac{d}{d x} [4]$

Step 1.1.3.3

Multiply

$9$ by

$1$ .

$9 + \frac{d}{d x} [4]$

$9 + \frac{d}{d x} [4]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since

$4$ is constant with respect to

$x$ , the derivative of

$4$ with respect to

$x$ is

$0$ .

$9 + 0$

Step 1.1.4.2

Add

$9$ and

$0$ .

$9$

$9$

$9$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

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

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

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

$3$ .

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

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

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

Step 6

Replace all occurrences of

$u$ with

$9 x + 4$ .

$\frac{1}{27} {(9 x + 4)}^{3} + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$