Calculus Examples

$\int \frac{1}{x \sqrt{x^{2} - 4}} d x$

Step 1

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

$\int \frac{1}{x \sqrt{x^{2} - 2^{2}}} d x$

Step 2

The integral of $\frac{1}{x \sqrt{x^{2} - 2^{2}}}$ with respect to $x$ is $\frac{1}{2} arcsec (| \frac{x}{2} |)$

$\frac{1}{2} arcsec (| \frac{x}{2} |) + C$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{2}$ and $arcsec (| \frac{x}{2} |)$ .

$\frac{arcsec (| \frac{x}{2} |)}{2} + C$

Step 3.2

Rewrite $\frac{arcsec (| \frac{x}{2} |)}{2} + C$ as $\frac{arcsec (\frac{| x |}{2})}{2} + C$ .

$\frac{arcsec (\frac{| x |}{2})}{2} + C$

Step 3.3

Reorder terms.

$\frac{1}{2} arcsec (\frac{1}{2} | x |) + C$