Explanation Given Information: The provided integral is ∫ t csc 3 t cot 3 t d t . Formula used: Write the formula used in integration by parts method. ∫ p q d x = p ∫ q d x − ∫ p ′ ( ∫ q d x ) d x Here, p = p ( x ) and q = q ( x ) Write the formula of integral of csc u cot u . ∫ csc u cot u d u = − csc u + C Write the formula of integral of csc u . ∫ csc u d u = − ln | csc u + cot u | + C Write the formula for derivative of u n . d d x u n = n u n − 1 The provided integral is, ∫ t csc 3 t cot 3 t d t Use the formula of integration by parts to find integral of above expression. ∫ t csc 3 t cot 3 t d t = t ∫ csc 3 t cot 3 t d t − ∫ d d t t ( ∫ csc 3 t cot 3 t d t ) d t = t ∫ csc 3 t cot 3 t d t − ∫ 1 ( ∫ csc 3 t cot 3 t d t ) d t Here, p = t and q = csc 3 t cot 3 t Multiply and divide by 3 . ∫ t csc 3 t cot 3 t d t = 1 3 t ∫ ( csc 3 t cot 3 t ) 3 d t ︸ I − 1 3 ∫ 1 ( ∫ ( csc 3 t cot 3 t ) 3 d t ) d t ︸ I I Again, multiply and divide by 3 in part II. ∫ t csc 3 t cot 3 t d t = 1 3 t ∫ ( csc 3 t cot 3 t ) 3 d t − 1 9 ∫ 1 ( ∫ ( csc 3 t cot 3 t ) 3 d t ) 3 d t let u = 3 t

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Chapter 8.5, Problem 37E

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To calculate: The integral of ∫tcsc3tcot3tdt.

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Chapter 8.5, Problem 36E

Chapter 8.5, Problem 38E

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Ch. 8.1 - For each angle, find a coterminal angle such that...Ch. 8.1 - Prob. 2CP Ch. 8.1 - Prob. 3CP Ch. 8.1 - Prob. 4CP Ch. 8.1 - Prob. 1SWU Ch. 8.1 - Prob. 2SWU Ch. 8.1 - Prob. 3SWU Ch. 8.1 - Prob. 4SWU Ch. 8.1 - Prob. 5SWU Ch. 8.1 - Prob. 6SWU

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The integral of ∫ t csc 3 t cot 3 t d t .

Solution Summary: The author explains the formula used in integration by parts method.

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To calculate: The integral of ∫tcsc3tcot3tdt.

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Chapter 8 Solutions