Calculus Examples

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

Step 1

Write $\frac{1}{x \sqrt{4 x^{2} - 1}}$ as a function.

$f (x) = \frac{1}{x \sqrt{4 x^{2} - 1}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

Simplify with factoring out.

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Step 4.1

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

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

Step 4.2

Rewrite $1$ as $1^{2}$ .

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

Step 5

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

$arcsec (| 2 x |) + C$

Step 6

Rewrite $arcsec (| 2 x |) + C$ as $arcsec (2 | x |) + C$ .

$arcsec (2 | x |) + C$

Step 7

The answer is the antiderivative of the function $f (x) = \frac{1}{x \sqrt{4 x^{2} - 1}}$ .

$F (x) =$ $arcsec (2 | x |) + C$