∫ (1 / (x√(x² - a²))) dx = (1/a) sec⁻¹(x/a) + c

Steps to Solve

1. Identify the integral form

We start with the integral

$$ \int \frac{1}{x \sqrt{x^2 - a^2}} , dx $$

This requires recognizing how we can manipulate the expression to facilitate integration.

2. Use substitution

We use the substitution

$$ x = a \sec(\theta) $$

This leads to

$$ dx = a \sec(\theta) \tan(\theta) , d\theta $$

3. Substitute in the integral

Now, substituting into the integral we get

$$ \int \frac{1}{a \sec(\theta) \sqrt{(a \sec(\theta))^2 - a^2}} \cdot a \sec(\theta) \tan(\theta) , d\theta $$

This simplifies since

$$ (a \sec(\theta))^2 - a^2 = a^2(\sec^2(\theta) - 1) = a^2 \tan^2(\theta) $$

4. Simplify the integrand

The integral now looks like

$$ \int \frac{a \sec(\theta) \tan(\theta)}{a \sec(\theta) \sqrt{a^2 \tan^2(\theta)}} , d\theta $$

This further simplifies to

$$ \int d\theta $$

5. Integrate

Integrating gives

$$ \theta + C $$

6. Back substitute

Recall our substitution

$$ \theta = \sec^{-1}\left(\frac{x}{a}\right) $$

Therefore, the final answer becomes

$$ \frac{1}{a} \sec^{-1}\left(\frac{x}{a}\right) + C $$