(i) `int (dx)/(x sqrt(x^2-1))` (ii) `int (dx)/(x sqrt(ax -x^2))`

Answer

Let's solve the given integrals step by step.

### Part (i): \(\int \frac{dx}{x \sqrt{x^2 - 1}}\)

**Step 1: Substitution**
Let \( x = \sec \theta \). Then, we have:
\[ dx = \sec \theta \tan \theta \, d\theta \]

**Step 2: Substitute in the integral**
Substituting \( x \) and \( dx \) into the integral:
\[ \int \frac{dx}{x \sqrt{x^2 - 1}} = \int \frac{\sec \theta \tan \theta \, d\theta}{\sec \theta \sqrt{\sec^2 \theta - 1}} \]
Since \( \sec^2 \theta - 1 = \tan^2 \theta \), we have:
\[ \sqrt{\sec^2 \theta - 1} = \tan \theta \]
Thus, the integral simplifies to:
\[ \int \frac{\sec \theta \tan \theta \, d\theta}{\sec \theta \tan \theta} = \int d\theta \]

**Step 3: Integrate**
The integral of \( d\theta \) is:
\[ \theta + C \]

**Step 4: Back substitute**
Since \( \theta = \sec^{-1}(x) \), we have:
\[ \int \frac{dx}{x \sqrt{x^2 - 1}} = \sec^{-1}(x) + C \]

### Part (ii): \(\int \frac{dx}{x \sqrt{ax - x^2}}\)

**Step 1: Substitution**
Let \( x = a \sin^2 \theta \). Then, we have:
\[ dx = 2a \sin \theta \cos \theta \, d\theta \]

**Step 2: Substitute in the integral**
Substituting \( x \) and \( dx \) into the integral:
\[ \int \frac{dx}{x \sqrt{ax - x^2}} = \int \frac{2a \sin \theta \cos \theta \, d\theta}{a \sin^2 \theta \sqrt{a \sin^2 \theta - a^2 \sin^4 \theta}} \]
This simplifies to:
\[ \int \frac{2 \sin \theta \cos \theta \, d\theta}{\sin^2 \theta \sqrt{a(1 - \sin^2 \theta)}} \]
Since \( \sqrt{1 - \sin^2 \theta} = \cos \theta \), we have:
\[ \int \frac{2 \sin \theta \cos \theta \, d\theta}{\sin^2 \theta \cdot \sqrt{a} \cos \theta} = \int \frac{2 \, d\theta}{\sqrt{a} \sin \theta} \]

**Step 3: Integrate**
The integral becomes:
\[ \frac{2}{\sqrt{a}} \int \csc \theta \, d\theta = \frac{2}{\sqrt{a}} (-\ln |\csc \theta + \cot \theta|) + C \]

**Step 4: Back substitute**
Using \( \sin^2 \theta = \frac{x}{a} \), we can express \( \csc \theta \) and \( \cot \theta \) in terms of \( x \):
\[ \csc \theta = \frac{\sqrt{a}}{\sqrt{x}}, \quad \cot \theta = \frac{\sqrt{a - x}}{\sqrt{x}} \]
Thus, the final result is:
\[ \int \frac{dx}{x \sqrt{ax - x^2}} = -\frac{2}{\sqrt{a}} \ln \left| \frac{\sqrt{a}}{\sqrt{x}} + \frac{\sqrt{a - x}}{\sqrt{x}} \right| + C \]

### Summary of Solutions:
1. \(\int \frac{dx}{x \sqrt{x^2 - 1}} = \sec^{-1}(x) + C\)
2. \(\int \frac{dx}{x \sqrt{ax - x^2}} = -\frac{2}{\sqrt{a}} \ln \left| \frac{\sqrt{a}}{\sqrt{x}} + \frac{\sqrt{a - x}}{\sqrt{x}} \right| + C\)