(i) `int sin^-1 x dx`
(ii) `int cos^-1 x dx`
(iii) `int cot^-1 x dx`

Answer

Let's solve the integrals step by step.

### (i) Integration of \( \sin^{-1} x \, dx \)

1. **Use Integration by Parts**:
\[ \int \sin^{-1} x \, dx = x \sin^{-1} x - \int x \cdot \frac{d}{dx}(\sin^{-1} x) \, dx \]
Here, we let \( u = \sin^{-1} x \) and \( dv = dx \).

2. **Differentiate \( \sin^{-1} x \)**:
\[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} \]

3. **Substitute into the equation**:
\[ \int \sin^{-1} x \, dx = x \sin^{-1} x - \int x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx \]

4. **Let \( t = 1 - x^2 \)**, then \( dt = -2x \, dx \) or \( x \, dx = -\frac{1}{2} dt \):
\[ \int x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx = -\frac{1}{2} \int \frac{1}{\sqrt{t}} \, dt \]

5. **Integrate**:
\[ -\frac{1}{2} \int t^{-1/2} \, dt = -\frac{1}{2} \cdot 2\sqrt{t} = -\sqrt{1 - x^2} \]

6. **Combine results**:
\[ \int \sin^{-1} x \, dx = x \sin^{-1} x + \sqrt{1 - x^2} + C \]

### (ii) Integration of \( \cos^{-1} x \, dx \)

1. **Use Integration by Parts**:
\[ \int \cos^{-1} x \, dx = x \cos^{-1} x - \int x \cdot \frac{d}{dx}(\cos^{-1} x) \, dx \]

2. **Differentiate \( \cos^{-1} x \)**:
\[ \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \]

3. **Substitute into the equation**:
\[ \int \cos^{-1} x \, dx = x \cos^{-1} x + \int \frac{x}{\sqrt{1 - x^2}} \, dx \]

4. **Let \( t = 1 - x^2 \)**, then \( dt = -2x \, dx \):
\[ \int \frac{x}{\sqrt{1 - x^2}} \, dx = -\frac{1}{2} \int \frac{1}{\sqrt{t}} \, dt \]

5. **Integrate**:
\[ -\frac{1}{2} \cdot 2\sqrt{t} = -\sqrt{1 - x^2} \]

6. **Combine results**:
\[ \int \cos^{-1} x \, dx = x \cos^{-1} x - \sqrt{1 - x^2} + C \]

### (iii) Integration of \( \cot^{-1} x \, dx \)

1. **Use Integration by Parts**:
\[ \int \cot^{-1} x \, dx = x \cot^{-1} x - \int x \cdot \frac{d}{dx}(\cot^{-1} x) \, dx \]

2. **Differentiate \( \cot^{-1} x \)**:
\[ \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2} \]

3. **Substitute into the equation**:
\[ \int \cot^{-1} x \, dx = x \cot^{-1} x + \int \frac{x}{1 + x^2} \, dx \]

4. **Let \( t = 1 + x^2 \)**, then \( dt = 2x \, dx \):
\[ \int \frac{x}{1 + x^2} \, dx = \frac{1}{2} \int \frac{1}{t} \, dt \]

5. **Integrate**:
\[ \frac{1}{2} \ln |t| = \frac{1}{2} \ln(1 + x^2) \]

6. **Combine results**:
\[ \int \cot^{-1} x \, dx = x \cot^{-1} x + \frac{1}{2} \ln(1 + x^2) + C \]

### Summary of Results:
1. \( \int \sin^{-1} x \, dx = x \sin^{-1} x + \sqrt{1 - x^2} + C \)
2. \( \int \cos^{-1} x \, dx = x \cos^{-1} x - \sqrt{1 - x^2} + C \)
3. \( \int \cot^{-1} x \, dx = x \cot^{-1} x + \frac{1}{2} \ln(1 + x^2) + C \)