If f and F are functions of x such that F′(x) = f(x) then the function F is called a primitive or an indefinite integral of f with respect to x and is expressed as
where c is a constant of integration.
f(x) then possesses infinitely many antiderivatives, all of them being contained in the expression F(x) + c where c is a constant.
From the geometric point of view, an indefinite integral is a collection (family) of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards.
x = a sin θ or a cos θ
x = a tan θ or a cot θ
x = a sec θ or a cosec θ
x = a cos θ or a cos 2 θ
x = a (1 - cos θ)