Calculus Examples

$\int \frac{1}{{(8 x - 1)}^{3}} d x$

Step 1

Let $u = 8 x - 1$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 8 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $8 x - 1$ .

$\frac{d}{d x} [8 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $8 x - 1$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.1.3.2

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

$8 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$8 + 0$

Step 1.1.4.2

Add $8$ and $0$ .

$8$

Step 1.2

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

$\int \frac{1}{u^{3}} \cdot \frac{1}{8} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{8}$ .

$\int \frac{1}{u^{3} \cdot 8} d u$

Step 2.2

Move $8$ to the left of $u^{3}$ .

$\int \frac{1}{8 u^{3}} d u$

Step 3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int \frac{1}{u^{3}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{8} \int {(u^{3})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{8} \int u^{3 \cdot - 1} d u$

Step 4.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{8} \int u^{- 3} d u$

Step 5

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

$\frac{1}{8} (- \frac{1}{2} u^{- 2} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{8} (- \frac{1}{2} u^{- 2} + C)$ as $\frac{1}{8} (- \frac{1}{2} \cdot \frac{1}{u^{2}}) + C$ .

$\frac{1}{8} (- \frac{1}{2} \cdot \frac{1}{u^{2}}) + C$

Step 6.2

Rewrite $\frac{1}{8} (- \frac{1}{2} \cdot \frac{1}{u^{2}}) + C$ as $\frac{1}{8} (- \frac{1}{u^{2} \cdot 2}) + C$ .

$\frac{1}{8} (- \frac{1}{u^{2} \cdot 2}) + C$

Step 6.3

Simplify.

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Step 6.3.1

Move $2$ to the left of $u^{2}$ .

$\frac{1}{8} (- \frac{1}{2 \cdot u^{2}}) + C$

Step 6.3.2

Multiply $\frac{1}{8}$ by $\frac{1}{2 u^{2}}$ .

$- \frac{1}{8 (2 u^{2})} + C$

Step 6.3.3

Multiply $2$ by $8$ .

$- \frac{1}{16 u^{2}} + C$

Step 7

Replace all occurrences of $u$ with $8 x - 1$ .

$- \frac{1}{16 {(8 x - 1)}^{2}} + C$