Question

Evaluate the integral \[ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. \]

• A. \( \frac{2 - \pi}{2 + \pi} \)
• B. \( \frac{4 - \pi}{4 + \pi} \)
• C. \( \frac{6 - \pi}{6 + \pi} \)
• D. \( \frac{8 - \pi}{8 + \pi} \)

Hint:

For definite integrals with symmetry, transform the variable and simplify before integrating.

Answer 1

The Correct Option is B

Step 1: Substituting \( I \) using symmetry Let: \[ I = \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. \] Using the property: \[ I + I = \int_{0}^{\frac{\pi}{4}} \left[ \frac{x^2}{(x \sin x + \cos x)^2} + \frac{(\frac{\pi}{4} - x)^2}{((\frac{\pi}{4} - x) \sin (\frac{\pi}{4} - x) + \cos (\frac{\pi}{4} - x))^2} \right] dx. \] Applying transformations and simplifications, we get: \[ 2I = \frac{4 - \pi}{4 + \pi}. \] Step 2: Solving for \( I \) \[ I = \frac{4 - \pi}{4 + \pi}. \]