Explanation 1) Concept: i) The substitution rule for definite integral: If g ' ( x ) is a continuous function on a , b whose f is continuous on range of u = g ( x ) , then ∫ a b f g x g ' ( x ) d x = ∫ g ( a ) g ( b ) f ( u ) d u . Here, g ( x ) is substituted as u and g ’ ( x ) d x = d u . ii) ∫ csc 2 ⁡ x d x = - c o t x + C 2) Given: ∫ π / 3 2 π / 3 csc 2 ⁡ 1 2 t d t 3) Calculation: The given integral is ∫ π / 3 2 π / 3 csc 2 ⁡ 1 2 t d t Here, use the substitution method. Substitute 1 2 t = u s o t = 2 u . Differentiating with respect to t d t = 2 d u The limits change and the new limits of integration are calculated by substituting for t = π 3 , u = 1 2 π 3 = π 6 & for t = 2 π 3 , u = <

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Chapter 4.5, Problem 40E

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The given definite integral ∫π/32π/3csc2⁡12tdt.

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Chapter 4.5, Problem 39E

Chapter 4.5, Problem 41E

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The given definite integral ∫ π / 3 2 π / 3 csc 2 ⁡ 1 2 t d t .

Solution Summary: The author explains the substitution rule for the given definite integral.

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To evaluate:

The given definite integral ∫π/32π/3csc2⁡12tdt.

Want to see the full answer?

Chapter 4 Solutions

The given definite integral ∫ π / 3 2 π / 3 csc 2 ⁡ 1 2 t d t .

Solution Summary: The author explains the substitution rule for the given definite integral.

Concept explainers

Videos

To evaluate: The given definite integral ∫π/32π/3csc2⁡12tdt.

Want to see the full answer?

Chapter 4 Solutions

To evaluate:

The given definite integral ∫π/32π/3csc2⁡12tdt.