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?
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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
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