What is the definite integral of 3csc^2(2x)dx from pi/8 to pi/4
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What is the definite integral of 3csc^2(2x)dx from pi/8 to pi/4

[From: ] [author: ] [Date: 12-10-13] [Hit: ]
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What is the definite integral of 3csc^2(2x)dx? I got -3/2 but the answer should be 3/2. Can you show me the solution?

-
3 ∫ csc^2 (2x) dx

u = 2x

du = 2 dx

3/2 ∫ csc^2 (u)

∫ csc^2(u) = -cot(u)

3/2 * - cot(u) = 3/2 * -cot(2x)

distribute the minus, which makes the right side positive
3/2 * -cot(2pi/4) - 3/2 * -cot(2pi/8)

cotangent of pi/2 is 0/1 = 0, cotangent of pi/4 = 1
3/2 * - cot(pi/2) + 3/2 cot(pi/4)

3/2 * 0 + 3/2 * 1

3/2
1
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