Calculus Examples

\int {(2 x - 3)}^{2} d x

$\int {(2 x - 3)}^{2} d x$

Step 1

Let

u = 2 x - 3

$u = 2 x - 3$ . Then

d u = 2 d x

$d u = 2 d x$ , so

\frac{1}{2} d u = d x

$\frac{1}{2} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = 2 x - 3

$u = 2 x - 3$ . Find

\frac{d u}{d x}

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

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Step 1.1.1

Differentiate

2 x - 3

$2 x - 3$ .

\frac{d}{d x} [2 x - 3]

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

Step 1.1.2

By the Sum Rule, the derivative of

2 x - 3

$2 x - 3$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.3

Evaluate

\frac{d}{d x} [2 x]

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

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Step 1.1.3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x

$2 x$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.1.3.3

Multiply

2

$2$ by

1

$1$ .

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

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

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

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3

$- 3$ with respect to

x

$x$ is

0

$0$ .

2 + 0

$2 + 0$

Step 1.1.4.2

Add

2

$2$ and

0

$0$ .

2

$2$

2

$2$

2

$2$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 2

Combine

u^{2}

$u^{2}$ and

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 3

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

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

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

Step 4

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

$\frac{1}{3} u^{3}$ .

\frac{1}{2} (\frac{1}{3} u^{3} + C)

$\frac{1}{2} (\frac{1}{3} u^{3} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite

\frac{1}{2} (\frac{1}{3} u^{3} + C)

$\frac{1}{2} (\frac{1}{3} u^{3} + C)$ as

\frac{1}{2} \cdot \frac{1}{3} u^{3} + C

$\frac{1}{2} \cdot \frac{1}{3} u^{3} + C$ .

\frac{1}{2} \cdot \frac{1}{3} u^{3} + C

$\frac{1}{2} \cdot \frac{1}{3} u^{3} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{1}{3}

$\frac{1}{3}$ .

\frac{1}{2 \cdot 3} u^{3} + C

$\frac{1}{2 \cdot 3} u^{3} + C$

Step 5.2.2

Multiply

2

$2$ by

3

$3$ .

\frac{1}{6} u^{3} + C

$\frac{1}{6} u^{3} + C$

\frac{1}{6} u^{3} + C

$\frac{1}{6} u^{3} + C$

\frac{1}{6} u^{3} + C

$\frac{1}{6} u^{3} + C$

Step 6

Replace all occurrences of

u

$u$ with

2 x - 3

$2 x - 3$ .

\frac{1}{6} {(2 x - 3)}^{3} + C

$\frac{1}{6} {(2 x - 3)}^{3} + C$

\int_{}^{} {(2 x - 3)}^{2} d x

$\int_{}^{} {(2 x - 3)}^{2} d x$

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

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