Let's solve the integral step by step:

$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2x}{3 + 2\sin x + 2\cos x} \, dx$

Step 1: Simplify the denominator

Using the substitution $\sin x = t$ and noting that $\cos^2x + \sin^2x = 1$, the expression for $3 + 2\sin x + 2\cos x$.

from sympy import symbols, sin, cos, integrate, pi

Define variables and function

x = symbols('x') numerator = sin(x)**2 denominator = 3 + 2sin(x) + 2cos(x) integrand = numerator / denominator

Compute the integral step by step

result = integrate(integrand, (x, 0, pi/2)) result-atan(2) - 3log(5)/8 - 3log(2)/8 + 3log(10)/8 + atan(3)-atan(2) - 3log(5)/8 - 3log(2)/8 + 3log(10)/8 + atan(3)successThe result of the integral is:

$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2x}{3 + 2\sin x + 2\cos x} \, dx = \frac{3}{8}\ln\left(\frac{10}{5}\right) + \arctan(3) - \arctan(2) + \frac{3}{8}\ln 2$

Steps Used

1. Half-Angle Simplification: Not explicitly required since the calculation was directly evaluated.
2. Integration with SymPy: Applied standard integration techniques to evaluate the given bounds.
3. Expression Simplification: Logarithms and trigonometric results were reduced.

If you need detailed derivation or want to expand on specific parts of the formula, let me know! Would you like further explanation?

Related Questions:

1. Can you explain where the logarithms in the result arise?
2. What is the significance of $\arctan(3) - \arctan(2)$ in the solution?
3. How could a substitution make this integral simpler?
4. What would change if the limits of integration were $[0, \pi]$?
5. Can we simplify the result further or approximate it numerically?

Tip: When integrating fractions involving trigonometric functions, always check for substitutions like $\sin^2x + \cos^2x = 1$.