`(i) int(cos^(3) x+ sin^(3) x)/(sin^(2) x.cos ^(2) x)dx " "(ii) int(cos2x)/(cos^(2) x sin^(2)x) dx`

Answer

Let's solve the given integrals step by step.

### Part (i):
Evaluate the integral
\[ I_1 = \int \frac{\cos^3 x + \sin^3 x}{\sin^2 x \cos^2 x} \, dx \]

**Step 1: Rewrite the numerator using the identity for cubes.**
\[ \cos^3 x + \sin^3 x = (\cos x + \sin x)(\cos^2 x - \cos x \sin x + \sin^2 x) \]
Since \(\cos^2 x + \sin^2 x = 1\), we can simplify:
\[ \cos^3 x + \sin^3 x = (\cos x + \sin x)(1 - \cos x \sin x) \]

**Step 2: Substitute this back into the integral.**
\[ I_1 = \int \frac{(\cos x + \sin x)(1 - \cos x \sin x)}{\sin^2 x \cos^2 x} \, dx \]

**Step 3: Split the integral into two parts.**
\[ I_1 = \int \frac{\cos x + \sin x}{\sin^2 x \cos^2 x} \, dx - \int \frac{(\cos x + \sin x) \cos x \sin x}{\sin^2 x \cos^2 x} \, dx \]

**Step 4: Simplify each integral.**
The first integral becomes:
\[ \int \frac{\cos x + \sin x}{\sin^2 x \cos^2 x} \, dx = \int \left( \frac{\cos x}{\sin^2 x \cos^2 x} + \frac{\sin x}{\sin^2 x \cos^2 x} \right) \, dx \]
This can be simplified to:
\[ \int \left( \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} \right) \, dx = \int \csc^2 x \, dx + \int \sec^2 x \, dx \]

**Step 5: Integrate.**
The integrals yield:
\[ -\cot x + \tan x + C \]

Thus, the final answer for part (i) is:
\[ I_1 = -\cot x + \tan x + C \]

### Part (ii):
Evaluate the integral
\[ I_2 = \int \frac{\cos 2x}{\cos^2 x \sin^2 x} \, dx \]

**Step 1: Use the double angle identity for cosine.**
\[ \cos 2x = \cos^2 x - \sin^2 x \]
Substituting this into the integral gives:
\[ I_2 = \int \frac{\cos^2 x - \sin^2 x}{\cos^2 x \sin^2 x} \, dx \]

**Step 2: Split the integral.**
\[ I_2 = \int \frac{\cos^2 x}{\cos^2 x \sin^2 x} \, dx - \int \frac{\sin^2 x}{\cos^2 x \sin^2 x} \, dx \]
This simplifies to:
\[ I_2 = \int \frac{1}{\sin^2 x} \, dx - \int \frac{1}{\cos^2 x} \, dx \]

**Step 3: Integrate each part.**
The integrals yield:
\[ -\cot x + \tan x + C \]

Thus, the final answer for part (ii) is:
\[ I_2 = -\cot x + \tan x + C \]

### Summary of Solutions:
1. For part (i):
\[ I_1 = -\cot x + \tan x + C \]
2. For part (ii):
\[ I_2 = -\cot x + \tan x + C \]