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1.55 Arcsine integral

Example.
01dx1-x2=π2.\int_{0}^{1}{{dx}\over{\sqrt{1-x^{2}}}}={{\pi}\over{2}}.

Note that 1/1-x21/\sqrt{1-x^{2}}\rightarrow\infty as x1-x\rightarrow 1-; hence the integral is improper.

Solution. We let x=sintx=\sin t, and observe that sint1-\sin t\rightarrow 1- as t(π/2)-t\rightarrow(\pi/2)-; so the limits 0<x<10<x<1 convert to: 0<t<π20<t<\frac{\pi}{2}.

Also, dx/dt=costdx/dt={\cos t} and 1-x2=1-sin2t=cost,\sqrt{1-x^{2}}={\sqrt{1-\sin^{2}t}}\,{=\cos t,} so

01dx1-x2=0π/2costcostdt=0π/2dt=π2.\int_{0}^{1}{{dx}\over{\sqrt{1-x^{2}}}}={\int_{0}^{\pi/2}{{\cos t}\over{\cos t% }}dt}\;{=\int_{0}^{\pi/2}dt}\;{={{\pi}\over{2}}.}