(i) `int (tanx)/((secx + tanx))dx` ,(ii) `int(cosecx)/((cosecx- cotx))dx`

Answer

Let's solve the two integration problems step by step.

### Problem (i):
\[ \int \frac{\tan x}{\sec x + \tan x} \, dx \]

**Step 1: Multiply by the conjugate**
We multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sec x - \tan x\):
\[ \int \frac{\tan x (\sec x - \tan x)}{(\sec x + \tan x)(\sec x - \tan x)} \, dx \]

**Step 2: Simplify the denominator**
Using the identity \((a+b)(a-b) = a^2 - b^2\), we simplify the denominator:
\[ \sec^2 x - \tan^2 x = 1 \]
Thus, the integral becomes:
\[ \int \frac{\tan x (\sec x - \tan x)}{1} \, dx = \int \tan x (\sec x - \tan x) \, dx \]

**Step 3: Expand the integrand**
Now we expand the integrand:
\[ \int (\tan x \sec x - \tan^2 x) \, dx \]

**Step 4: Separate the integral**
We can separate the integral into two parts:
\[ \int \tan x \sec x \, dx - \int \tan^2 x \, dx \]

**Step 5: Integrate each term**
1. The integral of \(\tan x \sec x\) is \(\sec x\).
2. The integral of \(\tan^2 x\) can be rewritten using the identity \(\tan^2 x = \sec^2 x - 1\):
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx = \tan x - x \]

Putting it all together:
\[ \int \tan x \sec x \, dx - \int \tan^2 x \, dx = \sec x - (\tan x - x) = \sec x - \tan x + x + C \]

### Final Answer for Problem (i):
\[ \int \frac{\tan x}{\sec x + \tan x} \, dx = \sec x - \tan x + x + C \]

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### Problem (ii):
\[ \int \frac{\csc x}{\csc x - \cot x} \, dx \]

**Step 1: Multiply by the conjugate**
We multiply the numerator and the denominator by the conjugate of the denominator, which is \(\csc x + \cot x\):
\[ \int \frac{\csc x (\csc x + \cot x)}{(\csc x - \cot x)(\csc x + \cot x)} \, dx \]

**Step 2: Simplify the denominator**
Using the identity \((a-b)(a+b) = a^2 - b^2\), we simplify the denominator:
\[ \csc^2 x - \cot^2 x = 1 \]
Thus, the integral becomes:
\[ \int \frac{\csc x (\csc x + \cot x)}{1} \, dx = \int \csc x (\csc x + \cot x) \, dx \]

**Step 3: Expand the integrand**
Now we expand the integrand:
\[ \int (\csc^2 x + \csc x \cot x) \, dx \]

**Step 4: Integrate each term**
1. The integral of \(\csc^2 x\) is \(-\cot x\).
2. The integral of \(\csc x \cot x\) is \(-\csc x\).

Putting it all together:
\[ \int \csc^2 x \, dx + \int \csc x \cot x \, dx = -\cot x - \csc x + C \]

### Final Answer for Problem (ii):
\[ \int \frac{\csc x}{\csc x - \cot x} \, dx = -\cot x - \csc x + C \]

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