Evaluate: (i) `intsecxlog(secx+tanx)\ dx` (ii) `intcos e c\ xlog(cos e c\ x-cotx)\ dx`

Answer

Let's solve the integrals step by step.

### Part (i): Evaluate \(\int \sec x \log(\sec x + \tan x) \, dx\)

**Step 1: Substitution**
Let \( t = \log(\sec x + \tan x) \).

**Hint:** When dealing with logarithmic integrals, substitution can simplify the expression.

**Step 2: Differentiate \( t \)**
Differentiating both sides gives:
\[ dt = \frac{1}{\sec x + \tan x} \left( \sec x \tan x + \sec^2 x \right) dx \]
This simplifies to:
\[ dt = \frac{\sec x (\tan x + \sec x)}{\sec x + \tan x} dx \]

**Hint:** Remember the derivatives of \(\sec x\) and \(\tan x\) for differentiation.

**Step 3: Rearranging**
We can express \( dx \) in terms of \( dt \):
\[ dx = \frac{(\sec x + \tan x)}{\sec x (\tan x + \sec x)} dt \]

**Step 4: Substitute back into the integral**
Now we can rewrite the integral:
\[ \int \sec x \log(\sec x + \tan x) \, dx = \int t \, dt \]

**Hint:** Look for patterns in the integral that allow for simplification.

**Step 5: Integrate**
Using the formula for the integral of \( t \):
\[ \int t \, dt = \frac{t^2}{2} + C \]

**Step 6: Substitute back for \( t \)**
Now substituting back for \( t \):
\[ \frac{(\log(\sec x + \tan x))^2}{2} + C \]

### Final Answer for Part (i):
\[ \int \sec x \log(\sec x + \tan x) \, dx = \frac{(\log(\sec x + \tan x))^2}{2} + C \]

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### Part (ii): Evaluate \(\int \csc x \log(\csc x - \cot x) \, dx\)

**Step 1: Substitution**
Let \( t = \log(\csc x - \cot x) \).

**Hint:** Similar to the first part, a logarithmic substitution can simplify the integral.

**Step 2: Differentiate \( t \)**
Differentiating gives:
\[ dt = \frac{1}{\csc x - \cot x} \left( -\csc x \cot x + \csc^2 x \right) dx \]
This simplifies to:
\[ dt = \frac{\csc^2 x - \csc x \cot x}{\csc x - \cot x} dx \]

**Hint:** Keep track of the signs when differentiating.

**Step 3: Rearranging**
Rearranging gives:
\[ dx = \frac{(\csc x - \cot x)}{(\csc^2 x - \csc x \cot x)} dt \]

**Step 4: Substitute back into the integral**
Now we can write:
\[ \int \csc x \log(\csc x - \cot x) \, dx = \int t \, dt \]

**Hint:** Recognize that the structure of the integral remains consistent.

**Step 5: Integrate**
Using the formula for the integral of \( t \):
\[ \int t \, dt = \frac{t^2}{2} + C \]

**Step 6: Substitute back for \( t \)**
Substituting back for \( t \):
\[ \frac{(\log(\csc x - \cot x))^2}{2} + C \]

### Final Answer for Part (ii):
\[ \int \csc x \log(\csc x - \cot x) \, dx = \frac{(\log(\csc x - \cot x))^2}{2} + C \]

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