GATE AG 2017 • Q29 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Limit, continuity and differentiability

Question Statement:

$I = \int \sqrt{a^2 - x^2} \, dx$ is

Options:

  • (A) $0.5[\sqrt{a^2 - x^2} + \sin^{-1}(x/a)]$
  • (B) $0.5[x\sqrt{a^2 - x^2} + \sin^{-1}(x/a)]$
  • (C) $0.5[\sqrt{a^2 - x^2} + a^2\sin^{-1}(x/a)]$
  • (D) $0.5[x\sqrt{a^2 - x^2} + a^2\sin^{-1}(x/a)]$

Official Answer Key:

D

Step-by-Step Solution & Explanation

• Standard result: ∫√(a²–x²) dx = (x/2)√(a²–x²) + (a²/2) sin⁻¹(x/a) + C. • This can be verified by differentiating the RHS and confirming it returns √(a²–x²). • Distractor logic: (A) and (C) drop the x multiplying the square-root term; (C) additionally is missing that x factor while keeping a² — only (D) has both terms correctly formed. • Final Answer: (D) Section 2 — Farm Machinery

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