- 2068 Shrawan
Integral of 1/(1+√tan x)
Short answer
Let I = ∫₀^{π/2} dx/(1+√tan x). Replace x by π/2−x to get I = ∫₀^{π/2} √sin x/(√sin x+√cos x) dx. Adding the two forms combines numerator and denominator, giving 2I = ∫₀^{π/2} dx = π/2, so I = π/4.
Evaluate: ∫₀^{π/2} dx/(1 + √tan x).
Let I = ∫₀^(π/2) dx/(1 + √tan x).
Write tan x = sin x/cos x, so 1 + √tan x = (√cos x + √sin x)/√cos x:
I = ∫₀^(π/2) √cos x/(√cos x + √sin x) dx
Replace x by π/2 − x (swap sin and cos):
I = ∫₀^(π/2) √sin x/(√sin x + √cos x) dx
Add the two forms:
2I = ∫₀^(π/2) (√cos x + √sin x)/(√sin x + √cos x) dx = ∫₀^(π/2) dx = π/2
I = π/4.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- DUIS: ∫ tan⁻¹(ax)/(x(1+x²)) dx · Mathematics I, chapter 2 · asked 2×
- Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0 · Mathematics I, chapter 4 · asked 6×
- Solve dy/dx = y/x + tan(y/x) · Mathematics I, chapter 4 · asked 6×
- Solve dy/dx − 2y tan x = y² tan x · Mathematics I, chapter 4 · asked 6×
- Solve (1+x²)dy/dx+y=e^(tan⁻¹x) · Mathematics I, chapter 4 · asked 6×
- Solve (1+y²)dx = (tan⁻¹y − x)dy · Mathematics I, chapter 4 · asked 2×
- Evaluate ∫∫ e^(x²) dx dy · Mathematics, chapter 2
- Evaluate ∫∫ e^(y/x) dy dx · Mathematics, chapter 2
- Evaluate a line integral F=sinyi+x(1+cosy)j · Mathematics III, chapter 2 · asked 6×
- Cauchy theorem vs formula; evaluate integral · Applied Mathematics, chapter 1 · asked 4×