Calculus Examples

$\int \frac{8 x}{{(4 x^{2} - 1)}^{4}} d x$

Step 1

Let $u = 4 x^{2} - 1$ . Then $d u = 8 x d x$ , so $\frac{1}{8 x} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x^{2} - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x^{2} - 1$ .

$\frac{d}{d x} [4 x^{2} - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$

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 $n = 2$ .

$4 (2 x) + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $2$ by $4$ .

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

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 $0$ .

$8 x + 0$

Step 1.1.4.2

Add $8 x$ and $0$ .

$8 x$

Step 1.2

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

$\int \frac{1}{u^{4}} d u$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $u^{4}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{4})}^{- 1} d u$

Step 2.2

Multiply the exponents in ${(u^{4})}^{- 1}$ .

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Step 2.2.1

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

$\int u^{4 \cdot - 1} d u$

Step 2.2.2

Multiply $4$ by $- 1$ .

$\int u^{- 4} d u$

Step 3

By the Power Rule, the integral of $u^{- 4}$ with respect to $u$ is $- \frac{1}{3} u^{- 3}$ .

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

Step 4

Simplify.

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Step 4.1

Rewrite $- \frac{1}{3} u^{- 3} + C$ as $- \frac{1}{3} \cdot \frac{1}{u^{3}} + C$ .

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 5

Replace all occurrences of $u$ with $4 x^{2} - 1$ .

$- \frac{1}{3 {(4 x^{2} - 1)}^{3}} + C$