Catalans constant

Calculus Level 3

∫ − π 2 π 2 x sin ⁡ ( x ) d x = a G \large \int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \frac{x}{\sin (x)} \, dx=a G

where G G is Catalan's constant . Submit a a .


The answer is 4.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Chew-Seong Cheong
Dec 19, 2017

I = ∫ − π 2 π 2 x sin ⁡ x d x Note that − x sin ⁡ ( − x ) = x sin ⁡ x = 2 ∫ 0 π 2 x sin ⁡ x d x See reference: G = 1 2 ∫ 0 π 2 x sin ⁡ x d x = 4 G where G is the Catalan’s constant. \begin{aligned} I & = \int_{-\frac \pi 2}^\frac \pi 2 \frac x{\sin x} dx & \small \color{#3D99F6} \text{Note that }\frac {-x}{\sin (-x)} = \frac x{\sin x} \\ & = 2\int_0^\frac \pi 2 \frac x{\sin x} dx & \small \color{#3D99F6} \text{See reference: }G = \frac 12 \int_0^\frac \pi 2 \frac x{\sin x} dx \\ & = 4G & \small \color{#3D99F6} \text{where }G \text{ is the Catalan's constant.} \end{aligned}

⟹ a = 4 \implies a = \boxed{4}

Reference: Catalan's constant

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