Evaluate this curious trigonometric integral in closed form

Calculus Level 4

Find the closed form of this integral and submit your answer as this number to 3 decimal places. $\int_0^{\pi/3} \dfrac{x - \sin (x)}{1 - \cos (x)} \csc\left( \frac x4 \right) \, dx$

The answer is 1.41915.

1 solution

Sangchul Lee
Mar 18, 2019

Mathematica gives $\int_{0}^{\pi/3} \frac{x-\sin(x)}{1-\cos(x)} \, \frac{\mathrm{d}x}{\sin(x/4)} = 2\sqrt{2+\sqrt{3}} - \frac{7\pi}{6\sqrt{2}} - \frac{\pi}{2\sqrt{6}} + \frac{\pi}{8}\log\left(15 - 10\sqrt{2} + 8\sqrt{3} - 6\sqrt{6}\right) + 6 \operatorname{Im}\big[ \chi_2(e^{i\pi/12}) \big],$ where $\chi_2$ is the Legendre chi-function of order 2. I will be surprised if this integral has an elementary closed form.

Yes sir, it has a beautiful and simple closed form. I dont use mathematica.

Srinivasa Raghava - 2 years, 2 months ago

I did half of computation by my hands, but eventually became lazy enough to throw the reduced expression into Mathematica and added some final touch to obtain the expression.

Anyway, is it really true that the expression $\operatorname{Im}\chi_2(e^{i\pi/12}) = \sum_{k=1,3,5,\cdots} \frac{\sin(k\pi/12)}{k^2}$ has an elementary closed form? At least I can see that this reduces to a linear combination of $\zeta(2)$ and the Dirichlet $L$ -functions for the characters from modulus 12, but I am not sure if we have simpler expression.

Sangchul Lee - 2 years, 2 months ago

What is the simple closed form? Please share.

Digvijay Singh - 2 years, 2 months ago

@Srinivasa Raghava Please tell how you arrived at the closed form???

Aaghaz Mahajan - 2 years, 2 months ago

Evaluate this curious trigonometric integral in closed form

The answer is 1.41915.

1 solution

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