Integral of √x/(√x+√(a−x))

Short answer

Apply ∫_0^a f(x)dx = ∫_0^a f(a-x)dx to write I = ∫_0^a √(a-x)/(√(a-x)+√x)dx. Adding the two forms yields 2I = ∫_0^a dx = a, so I = a/2.

Evaluate ∫₀^a √x/(√x + √(a−x)) dx.

Let I = ∫₀^a √x/(√x + √(a − x)) dx. Using ∫₀^a f(x) dx = ∫₀^a f(a − x) dx:

I = ∫₀^a √(a − x)/(√(a − x) + √x) dx

Add the two forms:

2I = ∫₀^a (√x + √(a − x))/(√x + √(a − x)) dx = ∫₀^a dx = a

Answer: I = a/2.

✓ Answer checked by Claude Opus 5.5 (high effort)

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