int_(0)^((pi)/(4))(sqrt(tan x))/(sin x cos x)dx=

int_(0)^((pi)/(4))sqrt(tan x)dx

int(sqrt(tan x))/(sin x cos x)dx=

int (sqrt (tan x)) / (sin x cos x) dx

Prove that: int_(0)^((pi)/(4))(sqrt(tan x)+sqrt(cot x)dx=sqrt(2)*(pi)/(2)

int (sqrt (tan x)) / (sin x cos x) * dx

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

int_ (0) ^ ((pi) / (2)) (sin x) / (sin x + cos x) dx = int_ (0) ^ ((pi) / (2)) (cos x) / (sin x + cos x) dx = int_ (0) ^ ((pi) / (2)) (dx) / (1 + cot x) = int_ (0) ^ ((pi) / (2)) (dx) / ( 1 + time x) = (pi) / (4)

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

Prove that: int_(0)^( pi/2)(sin x)/(sin x-cos x)dx=(pi)/(4)