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28. Integration of 1 / (x^5 + 1)

28. Integration of 1 / (x^5 + 1)

Question. (a) By using partial fractions, show that where (b) Hence, or otherwise, show that where (c) Hence, or otherwise, show that Solution. (a) (1) Factorisation of the denominator For the second factor, we seem to have two options: The first line gives while the second line yields Since the second line gives two complex conjugate roots, we select the first one and…

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27. Integration of 1 / (x^4 + 1)

27. Integration of 1 / (x^4 + 1)

Question. (a) By using partial fractions, show that (b) Hence, or otherwise, show that (c) Hence, or otherwise, show that Solution. (a) (1) Factorisation of the denominator: Can think of two possibilities: For real values of , we have to choose the first line with (the second line gives complex values for ): i.e. (2) Partial fractions Solving four simultaneous equations gives Finally, (b)…

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35. Integration of 1/(x^3+1)

35. Integration of 1/(x^3+1)

Question. (a) By using partial fractions, show that (b) Hence, or otherwise, show that (c) Hence, or otherwise, show that (a) (1) Factorisation of the denominator (2) By partial fractions, By the method of substitution (can equally use the method of equating the coefficients) which gives Hence, (b) Integration For the last part, we use the following substitution: which gives Finally, we…

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

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…

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28. Relations between trigonometric functions

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––––––

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Compositions of trigonometric and inverse trigonometric functions (35)

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.

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Compositions of trigonometric and inverse trigonometric functions (34)

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.

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

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…

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Compositions of trigonometric and inverse trigonometric functions (33)

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.

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Compositions of trigonometric and inverse trigonometric functions (32)

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.

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Compositions of trigonometric and inverse trigonometric functions (31)

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.

[mathsodology]

22. Integration of general powers of sine function (Generalisation of Wallis integral)

22. Integration of general powers of sine function (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…

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Compositions of trigonometric and inverse trigonometric functions (30)

Compositions of trigonometric and inverse trigonometric functions (30)

Proof of (30). Method 1. We can take the reciprocal of (6): 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.

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Compositions of trigonometric and inverse trigonometric functions (28)

Compositions of trigonometric and inverse trigonometric functions (28)

Proof of (28). Method 1. We can take the reciprocal of (4): 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.

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Compositions of trigonometric and inverse trigonometric functions (27)

Compositions of trigonometric and inverse trigonometric functions (27)

Proof of (27). Method 1. We can take the reciprocal of (3): 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.

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Compositions of trigonometric and inverse trigonometric functions (26)

Compositions of trigonometric and inverse trigonometric functions (26)

Proof of (26). Method 1. We can take the reciprocal of (2): 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.

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Compositions of trigonometric and inverse trigonometric functions (25)

Compositions of trigonometric and inverse trigonometric functions (25)

Proof of (25). Method 1. We can take the reciprocal of (1): 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.