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24. Integration of general powers of sine and cosine functions (feat. Euler beta and gamma functions)

Question 1. Let (a) By integration by parts where and , show that (b) By integration by parts where and , show that   Solution. (a) By integration by parts, we find (b) By integration by parts, we find Question 2. Let (a) By the substitution: , show that where the Euler beta function, also known as the Euler integral of the 1st kind, is defined by and the Euler gamma function, also known as…

28. Relations between trigonometric functions

1. Relations between trigonometric functions The inter-relations are obtained by the following identities: in terms of–––––– 2. Relations between the inverse trigonometric functions The following results are collected while working out compositions of trigonometric and inverse trigonometric functions. written as––––––

Compositions of trigonometric and inverse trigonometric functions (35)

Proof of (35). Method 1. We can take the reciprocal of (17): Method 2. Let where and , , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.

Compositions of trigonometric and inverse trigonometric functions (34)

(34) Method 1. We can take the reciprocal of (16): Method 2. Let where and , , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.

23. Integration of general powers of cosine function (A generalisation of Wallis integral)

Question. Let (a) By the substitution , show that where the Euler beta function, also known as the Euler integral of the 1st kind, is defined by and the Euler gamma function, also known as the Euler integral of the 2nd kind, is defined by Also, we note the following properties (see here for more details), (b) Using the recurrence relation, , show that where denotes the set of positive integers…

Compositions of trigonometric and inverse trigonometric functions (33)

Proof of (33). Method 1. We can take the reciprocal of (15): Method 2. Let where and , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.

Compositions of trigonometric and inverse trigonometric functions (32)

Proof of (32). Method 1. We can take the reciprocal of (14): Method 2. Let where and , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.

Compositions of trigonometric and inverse trigonometric functions (31)

Proof of (31). Method 1. We can take the reciprocal of (13): Method 2. Let where and , i.e. What we want is in terms of , so our strategy is to express in terms of using trigonometric identities.