Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Move the negative in front of the fraction.

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

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 6.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 6.2

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

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

Step 7

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

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

Step 8

The integral of $\frac{1}{\sqrt{1^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (u)$

$- (\arcsin (u) + C)$

Step 9

Simplify.

$- \arcsin (u) + C$

Step 10

Replace all occurrences of $u$ with $\ln (x)$ .

$- \arcsin (\ln (x)) + C$

Step 11

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

$F (x) =$ $- \arcsin (\ln (x)) + C$