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Evaluate the integral ∫_(π/4)^(π/2) (2 csc x)¹⁷ dx. Which of the following expressions represents its value?

1. ∫₀^(log(1+√(2))) 2(e^u + e^(-u))¹⁶ du
2. ∫₀^(log(1+√(2))) (e^u + e^(-u))¹⁷ du
3. ∫₀^(log(1+√(2))) (e^u - e^(-u))¹⁷ du
4. ∫₀^(log(1+√(2))) 2(e^u - e^(-u))¹⁶ du

Correct answer: ∫₀^(log(1+√(2))) 2(e^u + e^(-u))¹⁶ du

Solution

The integral is transformed using the substitution u = log(tan(x/2)), which converts the trigonometric expression into an exponential form. This leads to the equivalent integral ∫₀ˡᵒᵍ(1+√2) 2(e^u + e^(-u))¹⁶ du.

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