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

Steps to Solve

1. Rewrite the Integral Expression

We start with the integral

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

2. Substitution

Let ( x = a \sec(\theta) ). Then, the differential ( dx ) can be expressed as:

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

The square root ( \sqrt{x^2 - a^2} ) becomes:

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

3. Change Variables in the Integral

Substituting these into the integral gives:

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

which simplifies to:

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

4. Integrate

Now we can integrate:

$$ \int \frac{1}{a^2} , d\theta = \frac{\theta}{a^2} + C $$

5. Back Substitute for ( \theta )

Since we have ( \theta = \sec^{-1}\left(\frac{x}{a}\right) ), substituting back we find:

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

Now we can rewrite this as

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