(i) `int_0^pi (sin^2 x/2- cos^2 x/2) dx`
(ii) `int_0^(pi//2) (sin^2x)/(1+cosx)^2 dx`

Answer

To solve the given integrals step by step, we will break down each part of the question.

### Part (i): Evaluate the integral
\[ I = \int_0^\pi \left( \frac{\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}}{1} \right) dx \]

1. **Using the property of integrals**: We can use the property that states:
\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx \]
Here, let \( a = \pi \). Thus, we can write:
\[ I = \int_0^\pi \left( \sin^2 \frac{\pi - x}{2} - \cos^2 \frac{\pi - x}{2} \right) dx \]

2. **Simplifying the integrand**: We know that:
\[ \sin\left(\frac{\pi - x}{2}\right) = \cos\left(\frac{x}{2}\right) \quad \text{and} \quad \cos\left(\frac{\pi - x}{2}\right) = \sin\left(\frac{x}{2}\right) \]
Therefore, we can rewrite the integral as:
\[ I = \int_0^\pi \left( \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \right) dx \]

3. **Adding the two integrals**: Now, we can add the two expressions for \( I \):
\[ 2I = \int_0^\pi \left( \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} + \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} \right) dx \]
This simplifies to:
\[ 2I = \int_0^\pi 0 \, dx = 0 \]
Thus, we find:
\[ I = 0 \]

### Part (ii): Evaluate the integral
\[ I_2 = \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{(1 + \cos x)^2} \, dx \]

1. **Using the identity for \(\sin^2 x\)**: We can rewrite \(\sin^2 x\) as:
\[ \sin^2 x = 1 - \cos^2 x \]
Therefore, we can express the integral as:
\[ I_2 = \int_0^{\frac{\pi}{2}} \frac{1 - \cos^2 x}{(1 + \cos x)^2} \, dx \]

2. **Splitting the integral**: This can be split into two parts:
\[ I_2 = \int_0^{\frac{\pi}{2}} \frac{1}{(1 + \cos x)^2} \, dx - \int_0^{\frac{\pi}{2}} \frac{\cos^2 x}{(1 + \cos x)^2} \, dx \]

3. **Evaluating the first integral**: The first integral can be computed using the substitution \( u = \tan\left(\frac{x}{2}\right) \):
\[ \cos x = \frac{1 - u^2}{1 + u^2}, \quad dx = \frac{2}{1 + u^2} du \]
The limits change from \( x = 0 \) to \( x = \frac{\pi}{2} \) which corresponds to \( u = 0 \) to \( u = 1 \).

4. **Final evaluation**: After evaluating both parts and simplifying, we find:
\[ I_2 = 2 - \frac{\pi}{2} \]

### Final Answers:
1. For part (i): \( I = 0 \)
2. For part (ii): \( I_2 = 2 - \frac{\pi}{2} \)