The value of the integral \int_{0}^{\pi \/ 2} \sqrt{\sin x}/\sqrt{\sin x}+\sqrt{\cos x} d x is

NIMCET 2013 Question Paper with solutions

Question : 9 of 120

Marks: +1, -0

The value of the integral

∫↙{0}↖{π \/ 2} \;{√{\sin x}}/{√{\sin x}+√{\cos x}} d x

Solution:

Given

I=∫_{0}^{π/2}\;\;{√{\sinx}}/{√{\sinx}+√{\cosx}}dx

⇒I=∫_{0}^{π/2}\;\;{√{\sin(π/2-x)}}/{√{\sin(π/2-x)}+√{\cos(π/2-x)}}dx

⇒I=∫_{0}^{π/2}\;\;{√{\cosx}}/{√{\cosx}+√{\sinx}}dx

Now, adding Eqs. (i) and (ii), we get

2I=∫^{π/2}_{0}1dx⇒2I=[x]^{π/2}_{0}

⇒2I=π/2⇒I=π/4

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