If `alphain(0,pi/2)` then `sqrt(x^2+x)+(tanalpha)^2/(sqrt(x^2+x))` is always greater than or equal to

If alpha in(0,(pi)/(2)) then sqrt(x^(2)+x)+((tan alpha)^(2))/(sqrt(x^(2)+x)) is always greater than or equal to

If alpha in(0,(pi)/(2)), then sqrt(x^(2)+x)+(tan^(2)alpha)/(sqrt(x^(2)+x)) is always greater than or equal to (a)2tan alpha(b)1 (c) 2(d)sec2 alpha

Let a in (0,(pi)/(2)) and f(x)=sqrt(x^(2)+x)+(tan^(2)alpha)/(sqrt(x^(2)+x)), x gt 0 . If the least value of f(x) is 2sqrt3 , then alpha is equal to

int_(0)^((pi)/(2))(sqrt(cot x))/(sqrt(tan x)+sqrt(cot x))dx=

int_(0)^(pi//2) sqrt(sin x)/(sqrt(sin x ) + sqrt( cos x)) dx =