Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify the expression.

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Step 2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder terms.

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