In this paper, we derive equivalent equations of integration of secant and cosecant functions. Furthermore, we derive reduction formula for integration of product of integer powers of cosine and sine functions.

A Note on Integration of Trigonometric Functions

Introduction to real analysis

What is integration by parts

background

The author shows how ^-functions provide a natural setting in which to establish the change of variables formula for Lebesgue or Denjoy/Perron integrals. By abolishing the need to pass to the limit under the integral sign, the validity of the classical formula is significantly extended, yielding new results even in the case of Lebesgue integrals. 1. N-functions. A real-valued function on a finite interval is, by definition, an AZ-function if it carries null sets into null sets, the measure being Lebesgue measure (both here and in what follows). By a theorem of Banach ([1, p. 169]), any continuous A'-function possesses a finite derivative at each point of a set of positive measure. If this derivative, when extended by assigning it the value zero wherever it is not defined, happens to be Denjoy/Perron integrable, then its indefinite integral differs from the associated A'-function at most by a constant. This remarkable result is due to S. Saks [5, p. 145f.], extending Bary [2, p. 199f.], who had proved it for the Lebesgue integral. 2. The chain rule. Serrin and Varberg [7] have shown that a chain rule applies to the composition of two a.e. differentiable functions when the outer function is an A-function. A slight reformulation of their result brings out a useful extension property of the chain rule derivative. Extended chain rule. Let g be an a.e. differentiable real-valued function on the interval [a, b], and let F be an a.e. differentiable N-function defined on some interval which contains the range of g. Suppose that f is any function equivalent to F'. Then the function (f ° g)g' agrees with (F ° g)' a.e. on the set where the latter exists and vanishes a.e. on the remainder of [a, b]. Proof. Exactly the proof of Serrin and Varberg's Theorem 2, [7, p. 516], when their assumption of the a.e. differentiability of F ° g is dropped. 3. The change of variables formula. The foregoing considerations provide the basis for a transparent proof of a general Change of variables formula. Let g be a continuous, a.e. differentiable N-function on [a, b], and let F be a continuous, a.e differentiable N-function Received by the editors October 14, 1976. AMS (MOS) subject classifications (1970)'. Primary 26A39, 26A42; Secondary 26A30.

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A Note on Integration of Trigonometric Functions

Citations

Introduction to real analysis

What is integration by parts

References

Introduction to real analysis

What is integration by parts

Integration by substitution

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