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Calculus Examples

\int 6 {(2 x - 1)}^{- 3} d x

$\int 6 {(2 x - 1)}^{- 3} d x$ ,

u = 2 x - 1

$u = 2 x - 1$

Step 1

Let

u = 2 x - 1

$u = 2 x - 1$ . Then

d u = 2 d x

$d u = 2 d x$ . Rewrite using

u

$u$ and

d u

$d u$ .

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Step 1.1

Let

u = 2 x - 1

$u = 2 x - 1$ . Find

\frac{d u}{d x}

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

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Step 1.1.1

Differentiate

2 x - 1

$2 x - 1$ .

\frac{d}{d x} [2 x - 1]

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

Step 1.1.2

By the Sum Rule, the derivative of

2 x - 1

$2 x - 1$ with respect to

x

$x$ is

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

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

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

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

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} [- 1]

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

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} [- 1]

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

Step 1.1.3.3

Multiply

2

$2$ by

1

$1$ .

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

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

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

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

$x$ is

$0$ .

$2 + 0$

Step 1.1.4.2

Add

$2$ and

$0$ .

$2$

$2$

$2$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int \frac{3}{u^{3}} d u$

$\int \frac{3}{u^{3}} d u$

Step 2

Since

$3$ is constant with respect to

$u$ , move

$3$ out of the integral.

$3 \int \frac{1}{u^{3}} d u$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move

$u^{3}$ out of the denominator by raising it to the

$- 1$ power.

$3 \int {(u^{3})}^{- 1} d u$

Step 3.2

Multiply the exponents in

${(u^{3})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$3 \int u^{3 \cdot - 1} d u$

Step 3.2.2

Multiply

$3$ by

$- 1$ .

$3 \int u^{- 3} d u$

$3 \int u^{- 3} d u$

$3 \int u^{- 3} d u$

Step 4

By the Power Rule, the integral of

$u^{- 3}$ with respect to

$u$ is

$- \frac{1}{2} u^{- 2}$ .

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

Step 5

Simplify.

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Step 5.1

Simplify.

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Step 5.1.1

Combine

$u^{- 2}$ and

$\frac{1}{2}$ .

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

Step 5.1.2

Move

$u^{- 2}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 5.2

Simplify.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply

$- 1$ by

$3$ .

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

Step 5.3.2

Combine

$- 3$ and

$\frac{1}{2 u^{2}}$ .

$\frac{- 3}{2 u^{2}} + C$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{3}{2 u^{2}} + C$

$- \frac{3}{2 u^{2}} + C$

$- \frac{3}{2 u^{2}} + C$

Step 6

Replace all occurrences of

$u$ with

$2 x - 1$ .

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

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