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In order for the problem makes sense, it should be written in this precise form
if u= int (f(sin^2(x))sinx)dx on [0 ,pi/2] and v= int (f(cos^2(x))cosx)dx on [0 ,pi/2]
then u/v =......
Solution
Use the change of variables
u = int (f(sin^2(x))sinx)dx on [0 ,pi/2] = int (-f(sin^2(x))d(cos(x)) on [0 ,pi/2] = int (-f(1-cos^2(x))d(cos(x)) on [0 ,pi/2] =
= -int (f(1-t^2)dt on [1,0] (where t = cos(x) );
v = int (f(cos^2(x))cosx)dx [0 ,pi/2] = int (f(cos^2(x))d(sin(x))dx [0 ,pi/2] = int (f(1-sin^2(x))d(sin(x))dx [0 ,pi/2] =
= int (f(1-t^2)dt on [0,1] (where t = sin(x) ).
Since -int (f(1-t^2)dt on [1,0] = int (f(1-t^2)dt on [0,1], we have = 1.
Solved.