2 integration

Calculus Level 2

If ∫ 0 1 f ( x x + 1 − x ) d x = 10 \displaystyle \int_0^1 f \left(\frac {\sqrt x}{\sqrt x + \sqrt{1-x}} \right) dx = 10 , what is ∫ 0 1 f ( 1 − x x + 1 − x ) d x = ? \displaystyle \int_0^1 f \left(\frac {\sqrt{1-x}}{\sqrt x + \sqrt {1-x}} \right) dx = \ ?


The answer is 10.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Chew-Seong Cheong
Jul 19, 2020

Using the reflection identity : ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x \displaystyle \int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx , we have:

∫ 0 1 f ( x x + 1 − x ) d x = ∫ 0 1 f ( 1 − x 1 − x + x ) d x = 10 \int_0^1 f \left(\frac {\sqrt x}{\sqrt x + \sqrt{1-x}} \right) dx = \int_0^1 f \left(\frac {\sqrt{1-x}}{\sqrt {1-x} + \sqrt x} \right) dx = \boxed{10}

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