- 2071 Chaitra
Show ∫ dx/(x+√(a²−x²)) = π/4
Short answer
Put x = a sin θ, then I = ∫_0^{π/2} cosθ/(sinθ+cosθ)dθ. Using θ → π/2–θ gives the same integral with sinθ in numerator, adding gives 2I = π/2, hence I = π/4.
Show that ∫₀^a dx/(x + √(a²−x²)) = π/4.
Let I = ∫₀^a dx/(x + √(a² − x²)).
Put x = a sin θ, dx = a cos θ dθ, √(a² − x²) = a cos θ. Limits: x = 0 → θ = 0; x = a → θ = π/2.
I = ∫₀^(π/2) a cos θ dθ / (a sin θ + a cos θ) = ∫₀^(π/2) cos θ/(sin θ + cos θ) dθ
Replace θ by π/2 − θ: I = ∫₀^(π/2) sin θ/(cos θ + sin θ) dθ. Add:
2I = ∫₀^(π/2) (cos θ + sin θ)/(sin θ + cos θ) dθ = π/2
I = π/4. Shown.
✓ Answer checked by Claude Opus 5.5 (high effort)