Question

who know what is this transformation between cos(theta) and y and x?
the relation between cos(theta) and x and y?
please see this transformation at this URL:
http://uplod.ir/v2b9wab6a0x2/01.JPG.htm

Answer 1

this integration involves two variables (x' and r),this apostrophe means x dash ,so we will substitute each of them with theta and the constant distance y , cos(theta') = y / r ,by squaring, r^2 = y^2 * (sec(theta'))^2
tan(theta') = x' / y ,by differentiating both sides with respect to x',
(sec(theta'))^2 d(theta') = (dx')/y , therefore dx' = y * (sec(theta'))^2 * d(theta')

Answer 2

forgot to mention that theta' is angle between y and r' :)