Home page for accesible maths 1 1 Further Integration

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

1.55 Arcsine integral

Example.
∫01d⁢x1-x2=π2.\int_{0}^{1}{{dx}\over{\sqrt{1-x^{2}}}}={{\pi}\over{2}}.

Note that 1/1-x2→∞1/\sqrt{1-x^{2}}\rightarrow\infty as x→1-x\rightarrow 1-; hence the integral is improper.

Solution. We let x=sin⁡tx=\sin t, and observe that sin⁡t→1-\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, d⁢x/d⁢t=cos⁡tdx/dt={\cos t} and 1-x2=1-sin2⁡t=cos⁡t,\sqrt{1-x^{2}}={\sqrt{1-\sin^{2}t}}\,{=\cos t,} so

∫01d⁢x1-x2=∫0π/2cos⁡tcos⁡t⁢d⁢t=∫0π/2d⁢t=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}}.}