Integrability Formulas. Part III

For simplicity, we adopt the following convention: a, x denote real numbers, n denotes a natural number, A denotes a closed-interval subset of R, f , f1 denote partial functions from R to R, and Z denotes an open subset of R. One can prove the following propositions: (1) Suppose Z ⊆ dom((the function sec) · 1 idZ ). Then (i) −(the function sec) · 1 idZ is differentiable on Z, and (ii) for every x such that x ∈ Z holds (−(the function sec) · 1 idZ ) ′ Z(x) = (the function sin)( 1 x ) x2·(the function cos)( 1 x )2 . (2) Suppose Z ⊆ dom((the function cosec) · (the function exp)). Then (i) −(the function cosec) · (the function exp) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (−(the function cosec) · (the function exp))′ Z(x) = (the function exp)(x)·(the function cos)((the function exp)(x)) (the function sin)((the function exp)(x))2 . (3) Suppose Z ⊆ dom((the function cosec) · (the function ln)). Then (i) −(the function cosec) · (the function ln) is differentiable on Z, and