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Question Number 125034 by mathmax by abdo last updated on 07/Dec/20

find ∫_0 ^(π/2)  (x/(sinx))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{x}}{\mathrm{sinx}}\mathrm{dx} \\ $$

Commented by Olaf last updated on 07/Dec/20

The Catalan constant K has many  expressions. One of them is :  K = (1/2)∫_0 ^(π/2) (x/(sinx))dx  ⇒ ∫_0 ^(π/2) (x/(sinx))dx = 2K ≈ 1,831931188

$$\mathrm{The}\:\mathrm{Catalan}\:\mathrm{constant}\:\mathrm{K}\:\mathrm{has}\:\mathrm{many} \\ $$$$\mathrm{expressions}.\:\mathrm{One}\:\mathrm{of}\:\mathrm{them}\:\mathrm{is}\:: \\ $$$$\mathrm{K}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}}{\mathrm{sin}{x}}{dx} \\ $$$$\Rightarrow\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}}{\mathrm{sin}{x}}{dx}\:=\:\mathrm{2K}\:\approx\:\mathrm{1},\mathrm{831931188} \\ $$

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