---
title: "Show ∫ sin²x/(sin x+cos x) dx"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:integral-sin2x-over-sin-plus-cos"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-sin2x-over-sin-plus-cos
site: BCT Engineering
---

# Show ∫ sin²x/(sin x+cos x) dx

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2067 Asar
- Video: [90) show that Integral 0 to π/2 sin²x/sinx+cosx dx = 1/√2 log (√2+1)](https://www.youtube.com/watch?v=dcvnZnCT9Y8) — Sky Dhoni Production House
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-sin2x-over-sin-plus-cos
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Prove that ∫₀^{π/2} sin²x dx/(sin x + cos x) = (1/√2) log(√2+1).

## Short answer

Replace x by π/2−x and add to obtain 2I = ∫₀^{π/2} dx/(sin x+cos x). Write sin x+cos x = √2 sin(x+π/4), integrate csc to get 2I = (2/√2) ln(√2+1). Hence I = (1/√2) ln(√2+1).

## Answer

Let I = ∫₀^(π/2) sin²x/(sin x + cos x) dx. Replacing x by π/2 − x:

I = ∫₀^(π/2) cos²x/(sin x + cos x) dx

Add the two forms; since sin²x + cos²x = 1:

2I = ∫₀^(π/2) dx/(sin x + cos x) = K

**Evaluate K.** sin x + cos x = √2 sin(x + π/4), so

```text
K = (1/√2) ∫₀^(π/2) cosec(x + π/4) dx
  = (1/√2) [ln tan((x + π/4)/2)]₀^(π/2)
  = (1/√2) [ln tan(3π/8) − ln tan(π/8)]
  = (1/√2) ln[(√2+1)/(√2−1)]
  = (1/√2) · 2 ln(√2 + 1)
```
So 2I = (2/√2) ln(√2 + 1), and

**I = (1/√2) ln(√2 + 1).** Proved.

## Check yourself

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

1. Replacing x by π/2 - x gives
   A. integral sin^2 x/(sin x - cos x)
   B. integral cos^2 x/(sin x + cos x)
   C. integral cos x/(sin x + cos x)
   D. integral sin x/(sin^2 x + cos^2 x)
   Answer: B. sin and cos swap. (Key point: Second form)

2. Adding gives 2I = integral 1/(sin x + cos x) because
   A. sin x cos x = 1/2
   B. the denominators cancel
   C. sin^2 x + cos^2 x = 1
   D. sin x = cos x
   Answer: C. Numerators sum to 1. (Key point: Adding)

3. Integral of cosec(x + π/4) is evaluated as
   A. ln sec(x+π/4)
   B. -cot(x+π/4)
   C. ln sin(x + π/4)
   D. ln tan((x+π/4)/2)
   Answer: D. Standard integral of cosec. (Key point: Cosecant)

4. I equals
   A. (1/sqrt2) ln(sqrt2 + 1)
   B. sqrt2 ln(sqrt2+1)
   C. ln(sqrt2+1)
   D. (1/2) ln(sqrt2+1)
   Answer: A. 2I = (2/sqrt2) ln(sqrt2+1). (Key point: Result)

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