Several Integrability Formulas of Special Functions. Part II

Several Integrability Formulas of Special Functions. Part II In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3]. MML identifier: INTEGR11, version: 7.11.01 4.117.1046


Differentiation Formulas
For simplicity, we adopt the following rules: r, x, a, b denote real numbers, n, m denote elements of N, A denotes a closed-interval subset of R, and Z denotes an open subset of R.

Addenda
In the sequel f , f 1 , f 2 , f 3 , g are partial functions from R to R.
The following propositions are true: (37) Suppose Z ⊆ dom((the function tan)+(the function sec)) and for every x such that x ∈ Z holds 1 + sin x = 0 and 1 − sin x = 0. Then (i) (the function tan)+(the function sec) is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((the function tan)+(the function sec))  (v) f A is continuous. (i) (id Z + the function cot)−the function cosec is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((id Z +the function cot)−the function cosec) for every x such that x ∈ Z holds 1 + cos x = 0 and 1 − cos x = 0 and f (x) = cos x 1+cos x , (iii) dom((id Z + the function cot)−the function cosec) = Z, (51) Suppose Z ⊆ dom(id Z + the function cot+the function cosec) and for every x such that x ∈ Z holds 1 + cos x = 0 and 1 − cos x = 0. Then (i) id Z + the function cot+the function cosec is differentiable on Z, and (ii) for every x such that x ∈ Z holds (id Z + the function cot+the function cosec) for every x such that x ∈ Z holds 1 + cos x = 0 and 1 − cos x = 0 and f (x) = cos x cos x−1 , (iii) dom(id Z + the function cot+the function cosec) = Z, (iv) Z = dom f, and (v) f A is continuous.
(53) Suppose Z ⊆ dom((id Z − the function tan)+the function sec) and for every x such that x ∈ Z holds 1 + sin x = 0 and 1 − sin x = 0. Then (i) (id Z − the function tan)+the function sec is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((id Z −the function tan)+the function sec)

nuous. Then
The following propositions are true: for every x such that x ∈ Z holds sin x > 0, (iii) Z ⊆ dom((the function ln) ·(the function sin)), (iv) Z = dom (the function cot), and (v) (the function cot) A is continuous.
Then A (the function cot)(x)dx = ln sin sup A − ln sin inf A.