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int(0)^(pi//2)(1)/((1+sqrt(cotx)))dx=?...

`int_(0)^(pi//2)(1)/((1+sqrt(cotx)))dx=?`

A

`0`

B

`(pi)/(4)`

C

`(pi)/(2)`

D

`pi`

Text Solution

AI Generated Solution

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\cot x}} \, dx \), we will follow these steps: ### Step 1: Rewrite the Integral We start by rewriting the integrand. Recall that \( \cot x = \frac{\cos x}{\sin x} \), so we can express the integral as: \[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\frac{\cos x}{\sin x}}} \, dx \] ...
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