Calculus Examples

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

Step 1

Cancel the common factor of $2 x - 4$ and $2 x^{2} - 8$ .

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Step 1.1

Factor $2$ out of $2 x$ .

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

Step 1.2

Factor $2$ out of $- 4$ .

$\frac{d}{d x} [\int \frac{2 (x) + 2 \cdot - 2}{2 x^{2} - 8} d x]$

Step 1.3

Factor $2$ out of $2 (x) + 2 (- 2)$ .

$\frac{d}{d x} [\int \frac{2 (x - 2)}{2 x^{2} - 8} d x]$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 x^{2}$ .

$\frac{d}{d x} [\int \frac{2 (x - 2)}{2 (x^{2}) - 8} d x]$

Step 1.4.2

Factor $2$ out of $- 8$ .

$\frac{d}{d x} [\int \frac{2 (x - 2)}{2 x^{2} + 2 \cdot - 4} d x]$

Step 1.4.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

$\frac{d}{d x} [\int \frac{2 (x - 2)}{2 (x^{2} - 4)} d x]$

Step 1.4.4

Cancel the common factor.

$\frac{d}{d x} [\int \frac{2 (x - 2)}{2 (x^{2} - 4)} d x]$

Step 1.4.5

Rewrite the expression.

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

Step 2

$\int \frac{x - 2}{x^{2} - 4} d x$ is an antiderivative of $\frac{x - 2}{x^{2} - 4}$ , so by definition $\frac{d}{d x} [\int \frac{x - 2}{x^{2} - 4} d x]$ is $\frac{x - 2}{x^{2} - 4}$ .

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Step 2.1

Factor $2$ out of $2 x$ .

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

Step 2.2

Factor $2$ out of $- 4$ .

$\frac{2 (x) + 2 \cdot - 2}{2 x^{2} - 8}$

Step 2.3

Factor $2$ out of $2 (x) + 2 (- 2)$ .

$\frac{2 (x - 2)}{2 x^{2} - 8}$

Step 2.4

Cancel the common factors.

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Step 2.4.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x - 2)}{2 (x^{2}) - 8}$

Step 2.4.2

Factor $2$ out of $- 8$ .

$\frac{2 (x - 2)}{2 x^{2} + 2 \cdot - 4}$

Step 2.4.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 4$ .

$\frac{2 (x - 2)}{2 (x^{2} - 4)}$

Step 2.4.4

Cancel the common factor.

$\frac{2 (x - 2)}{2 (x^{2} - 4)}$

Step 2.4.5

Rewrite the expression.

$\frac{x - 2}{x^{2} - 4}$

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{x - 2}{x^{2} - 2^{2}}$

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x - 2}{(x + 2) (x - 2)}$

Step 3.2

Cancel the common factor of $x - 2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x - 2}{(x + 2) (x - 2)}$

Step 3.2.2

Rewrite the expression.

$\frac{1}{x + 2}$