Integral of $$\frac{1}{x\sqrt{x^2-a^2}}$$

To evaluate this symmetric integration problem , we divide the numerator and denominator by powers of $$x$$ to transform the expression into a standard functional form.

Step 1: Divide the numerator and denominator by $$x^2$$,we factor out $$x$$ from the first factor of the denominator, and $$x^2$$ from inside the radical of the second factor then we get:

$$I = \int \frac{1 - \frac{1}{x^2}}{\left(x + \frac{1}{x}\right) \sqrt{\left(x^2 + \frac{1}{x^2}\right)}} \, dx$$

Step 2: Express the variables using substitution

Let $$t = x + \frac{1}{x}$$. Differentiating both sides with respect to $$x$$ gives:

$$dt = \left(1 - \frac{1}{x^2}\right) dx$$

We also need to express the terms inside the radical in terms of $$t$$ by squaring our substitution variable:

$$t^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \implies x^2 + \frac{1}{x^2} = t^2 - 2$$

Substituting $$t = x + \frac{1}{x}$$ directly converts the expression into:

$$I = \int \frac{dt}{t\sqrt{t^2 - 2}}$$

Step 3: Evaluate the standard integral form

To evaluate $$\int \frac{dt}{t\sqrt{t^2 - 2}}$$, we can substitute $$t = \sqrt{2}\sec\theta$$, which gives:

$$dt = \sqrt{2}\sec\theta\tan\theta \, d\theta$$

Substitute these trigonometric terms into the integral expression:

$$I = \int \frac{\sqrt{2}\sec\theta\tan\theta \, d\theta}{\sqrt{2}\sec\theta\sqrt{2\sec^2\theta - 2}}$$

$$I = \int \frac{\sqrt{2}\sec\theta\tan\theta \, d\theta}{\sqrt{2}\sec\theta \cdot \sqrt{2}\tan\theta} = \int \frac{1}{\sqrt{2}} \, d\theta$$

Integrating a constant value gives:

$$I = \frac{1}{\sqrt{2}}\theta + C$$

Step 4: Substitute back to match the target form

Now, replace the temporary variable $$\theta$$ with our initial variable track expressions:

$$\sec\theta = \frac{t}{\sqrt{2}} \implies \theta = \sec^{-1}\left(\frac{t}{\sqrt{2}}\right)$$

Substitute $$t = x + \frac{1}{x} = \frac{x^2 + 1}{x}$$ back into the angle slot:

$$I = \frac{1}{\sqrt{2}}\sec^{-1}\left(\frac{x^2 + 1}{x\sqrt{2}}\right) + C$$

Comparing this directly with the given template in the question $$k \sec^{-1}\left(\frac{x^2 + 1}{x\sqrt{2}}\right) + C$$:

$$k= \frac{1}{\sqrt{2}}$$