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Question Number 83405 by jagoll last updated on 02/Mar/20

lim_(b→1^− )  ∫_0 ^b  ((sin (x))/(√(1−x^2 ))) dx ?

$$\underset{\mathrm{b}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{b}} \:\frac{\mathrm{sin}\:\left(\mathrm{x}\right)}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:?\: \\ $$

Commented by mr W last updated on 02/Mar/20

let x=sin θ  =∫_0 ^(π/2) sin (sin θ) dθ  =∫_0 ^(π/2) {2Σ_(k=0) ^∞ J_(2k+1) (1) sin (2k+1)θ}dθ  =−2[Σ_(k=0) ^∞ ((J_(2k+1) (1))/(2k+1)) cos (2k+1)θ]_0 ^(π/2)   =2Σ_(k=0) ^∞ ((J_(2k+1) (1))/(2k+1))

$${let}\:{x}=\mathrm{sin}\:\theta \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:\left(\mathrm{sin}\:\theta\right)\:{d}\theta \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left\{\mathrm{2}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{J}_{\mathrm{2}{k}+\mathrm{1}} \left(\mathrm{1}\right)\:\mathrm{sin}\:\left(\mathrm{2}{k}+\mathrm{1}\right)\theta\right\}{d}\theta \\ $$$$=−\mathrm{2}\left[\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{J}_{\mathrm{2}{k}+\mathrm{1}} \left(\mathrm{1}\right)}{\mathrm{2}{k}+\mathrm{1}}\:\mathrm{cos}\:\left(\mathrm{2}{k}+\mathrm{1}\right)\theta\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$=\mathrm{2}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{J}_{\mathrm{2}{k}+\mathrm{1}} \left(\mathrm{1}\right)}{\mathrm{2}{k}+\mathrm{1}} \\ $$

Commented by jagoll last updated on 02/Mar/20

thank you sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

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