Evaluate `int_0^(pi/4)(sqrt(tanx)+sqrt(cotx))dx`

Explanation

`I = int (sqrt(sin x/cos x) + sqrt(cos x/sinx) ) dx`

`= int( sinx + cosx)/(sqrtcosx sqrtsinx) dx`

`= int(sin x + cos x)/(sqrt(cos x sin x))dx`

now , `sin x + cosx `

integral is `- cosx + sinx`

now, doing `(sin x - cos x)^2`

`= sin^2x + cos^2 x - 2 sinx cos x`

`(sin x -cos x)^2 = 1- 2 sinxcosx`

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