GATE AG 2023 • Q13 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Homogeneous function – Euler's theorem on homogeneous functions

Question Statement:

The value of $I = \int_{0}^{\pi/2} \frac{(\sin x + \cos x)^2}{\sqrt{1+\sin 2x}} \, dx$ is:

Options:

  • (A) 0
  • (B) 1
  • (C) 2
  • (D) 3

Official Answer Key:

C

Step-by-Step Solution & Explanation

Expand (sinx+cosx)² = sin²x+cos²x+2sinxcosx = 1+sin2x. So the integrand becomes (1+sin2x)/√(1+sin2x) = √(1+sin2x) = √[(sinx+cosx)²] = |sinx+cosx|. On [0,π/2], sinx+cosx ≥ 0, so |sinx+cosx| = sinx+cosx. Thus I = ∫₀^(π/2)(sinx+cosx)dx = [−cosx+sinx]₀^(π/2) = (0+1)−(−1+0) = 1+1 = 2. Verified symbolically by computer algebra. Answer: (C) 2.

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