(i) `int_0^(pi//4) tan x dx` (ii) `int_(pi//4)^(pi//2) cot x dx`

Answer

To solve the integrals given in the question, we will break them down step by step.

### Part (i): Evaluate \( \int_0^{\frac{\pi}{4}} \tan x \, dx \)

1. **Rewrite the integral**:
\[ \tan x = \frac{\sin x}{\cos x} \]
So, we can rewrite the integral as:
\[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x}{\cos x} \, dx \]

2. **Substitution**:
Let \( t = \cos x \). Then, the derivative \( dt = -\sin x \, dx \) or \( \sin x \, dx = -dt \).
When \( x = 0 \), \( t = \cos(0) = 1 \).
When \( x = \frac{\pi}{4} \), \( t = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

3. **Change the limits and substitute**:
The integral becomes:
\[ I = \int_{1}^{\frac{1}{\sqrt{2}}} \frac{-1}{t} \, dt \]
This simplifies to:
\[ I = -\int_{1}^{\frac{1}{\sqrt{2}}} \frac{1}{t} \, dt \]

4. **Integrate**:
The integral of \( \frac{1}{t} \) is \( \log |t| \):
\[ I = -\left[ \log |t| \right]_{1}^{\frac{1}{\sqrt{2}}} \]
Evaluating this gives:
\[ I = -\left( \log\left(\frac{1}{\sqrt{2}}\right) - \log(1) \right) = -\left( -\frac{1}{2} \log(2) \right) = \frac{1}{2} \log(2) \]

### Part (ii): Evaluate \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot x \, dx \)

1. **Rewrite the integral**:
\[ \cot x = \frac{\cos x}{\sin x} \]
Thus, we can rewrite the integral as:
\[ J = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x}{\sin x} \, dx \]

2. **Substitution**:
Let \( t = \sin x \). Then, \( dt = \cos x \, dx \).
When \( x = \frac{\pi}{4} \), \( t = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).
When \( x = \frac{\pi}{2} \), \( t = \sin\left(\frac{\pi}{2}\right) = 1 \).

3. **Change the limits and substitute**:
The integral becomes:
\[ J = \int_{\frac{1}{\sqrt{2}}}^{1} \frac{1}{t} \, dt \]

4. **Integrate**:
The integral of \( \frac{1}{t} \) is \( \log |t| \):
\[ J = \left[ \log |t| \right]_{\frac{1}{\sqrt{2}}}^{1} \]
Evaluating this gives:
\[ J = \log(1) - \log\left(\frac{1}{\sqrt{2}}\right) = 0 + \frac{1}{2} \log(2) = \frac{1}{2} \log(2) \]

### Final Results
- For part (i):
\[ \int_0^{\frac{\pi}{4}} \tan x \, dx = \frac{1}{2} \log(2) \]
- For part (ii):
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot x \, dx = \frac{1}{2} \log(2) \]