---
title: "Show ∫ dx/(x+√(a²−x²)) = π/4"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:integral-dx-over-x-plus-sqrt"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-dx-over-x-plus-sqrt
site: BCT Engineering
---

# Show ∫ dx/(x+√(a²−x²)) = π/4

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2071 Chaitra
- Video: [Show that Integral 0 to a dx / x + √a² - x² = π/4 | Integration of dx / x + √a² - x²](https://www.youtube.com/watch?v=cNur7Zqvo30) — Nitin Kumar Gupta Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-dx-over-x-plus-sqrt
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Show that ∫₀^a 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.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The substitution used is
   A. x = a tan θ
   B. x = a sin θ
   C. x = a cos 2θ
   D. x = a θ
   Answer: B. Then sqrt(a^2-x^2) = a cos θ. (Key point: Substitution)

2. The integral becomes
   A. integral dθ/(sin θ + cos θ)
   B. integral sin θ/(sin θ + cos θ) dθ from 0 to π
   C. integral cos θ/(sin θ + cos θ) dθ from 0 to π/2
   D. integral cos θ dθ from 0 to π/2
   Answer: C. a cancels. (Key point: Reduced form)

3. Replacing θ by π/2 - θ gives
   A. cos θ/(sin θ - cos θ)
   B. sin θ cos θ
   C. 1/(sin θ + cos θ)
   D. sin θ/(sin θ + cos θ)
   Answer: D. Adding gives integrand 1. (Key point: Trick)

4. I equals
   A. π/4
   B. π/2
   C. π/8
   D. 1
   Answer: A. 2I = π/2. (Key point: Result)