(i) `int sec^3 x dx` (ii) `int cosec^3 x dx`

Answer

Let's solve the integrals step by step.

### Part (i): \(\int \sec^3 x \, dx\)

1. **Rewrite the integral**:
\[ \int \sec^3 x \, dx = \int \sec x \cdot \sec^2 x \, dx \]

2. **Use integration by parts**:
Let \( u = \sec x \) and \( dv = \sec^2 x \, dx \). Then, \( du = \sec x \tan x \, dx \) and \( v = \tan x \).

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int \sec^3 x \, dx = \sec x \tan x - \int \tan x \sec x \tan x \, dx \]

3. **Simplify the integral**:
\[ \int \sec^3 x \, dx = \sec x \tan x - \int \sec x \tan^2 x \, dx \]
Recall that \(\tan^2 x = \sec^2 x - 1\):
\[ \int \sec^3 x \, dx = \sec x \tan x - \int \sec x (\sec^2 x - 1) \, dx \]
This expands to:
\[ \int \sec^3 x \, dx = \sec x \tan x - \int \sec^3 x \, dx + \int \sec x \, dx \]

4. **Rearranging the equation**:
\[ 2 \int \sec^3 x \, dx = \sec x \tan x + \int \sec x \, dx \]
Thus:
\[ \int \sec^3 x \, dx = \frac{1}{2} \left( \sec x \tan x + \int \sec x \, dx \right) \]

5. **Integrate \(\int \sec x \, dx\)**:
The integral of \(\sec x\) is:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]

6. **Final result**:
Substituting back, we get:
\[ \int \sec^3 x \, dx = \frac{1}{2} \left( \sec x \tan x + \ln |\sec x + \tan x| \right) + C \]

### Part (ii): \(\int \csc^3 x \, dx\)

1. **Rewrite the integral**:
\[ \int \csc^3 x \, dx = \int \csc x \cdot \csc^2 x \, dx \]

2. **Use integration by parts**:
Let \( u = \csc x \) and \( dv = \csc^2 x \, dx \). Then, \( du = -\csc x \cot x \, dx \) and \( v = -\cot x \).

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \int \csc^3 x \, dx = -\csc x \cot x + \int \cot x \csc x \cot x \, dx \]

3. **Simplify the integral**:
\[ \int \csc^3 x \, dx = -\csc x \cot x + \int \csc x \cot^2 x \, dx \]
Recall that \(\cot^2 x = \csc^2 x - 1\):
\[ \int \csc^3 x \, dx = -\csc x \cot x + \int \csc x (\csc^2 x - 1) \, dx \]
This expands to:
\[ \int \csc^3 x \, dx = -\csc x \cot x + \int \csc^3 x \, dx - \int \csc x \, dx \]

4. **Rearranging the equation**:
\[ 2 \int \csc^3 x \, dx = -\csc x \cot x - \int \csc x \, dx \]
Thus:
\[ \int \csc^3 x \, dx = \frac{1}{2} \left( -\csc x \cot x - \int \csc x \, dx \right) \]

5. **Integrate \(\int \csc x \, dx\)**:
The integral of \(\csc x\) is:
\[ \int \csc x \, dx = -\ln |\csc x + \cot x| + C \]

6. **Final result**:
Substituting back, we get:
\[ \int \csc^3 x \, dx = -\frac{1}{2} \csc x \cot x + \frac{1}{2} \ln |\csc x + \cot x| + C \]