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Prove that int(0)^(a) f(x) dx = int(0)^(...

Prove that `int_(0)^(a) f(x) dx = int_(0)^(a) f(a - x)dx` and hence evaluate the following:
(c) `int_(0)^(pi/2)(sqrt(sinx))/(sqrt(sin x) + sqrt(cos x))dx`

Answer

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Knowledge Check

  • int_0^(pi//2)(sqrt(sin x))/(sqrt(sin x) + sqrt(cos x)) dx equals:

    A
    `pi/4`
    B
    `pi/2`
    C
    zero
    D
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  • int (sin 5x)/(sqrt(cos 5x)) dx =

    A
    `2/5 sqrt(cos 5x)`
    B
    `-2/5 sqrt(cos 5x)`
    C
    `10 sqrt(cos 5x)`
    D
    `-10 sqrt(cos 5x)`

  • int_0^(pi//2) (sqrt( cot x))/(sqrt(cot x) - sqrt(tan x)) dx is :

    A
    `pi//4`
    B
    `pi//2`
    C
    `pi`
    D
    `0`

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