Question Number 127904 by bramlexs22 last updated on 03/Jan/21
$$\:\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\mathrm{100}} \:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}\left(\mathrm{100}−\mathrm{x}\right)}}\:=\:\pi \\ $$
Answered by liberty last updated on 03/Jan/21
![let x = 100 sin^2 t ⇒dx= 200 sin t cos t dt ∫_0 ^( (π/2)) ((200 sin t cos t dt)/( (√(100sin^2 t(100 cos^2 t))))) = ∫_0 ^( (π/2)) 2 dt = 2t ]_0 ^(π/2) = 2×(π/2)=π](Q127907.png)
$$\:\mathrm{let}\:\mathrm{x}\:=\:\mathrm{100}\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{t}\:\Rightarrow\mathrm{dx}=\:\mathrm{200}\:\mathrm{sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{t}\:\mathrm{dt} \\ $$$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{200}\:\mathrm{sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{t}\:\mathrm{dt}}{\left.\:\sqrt{\mathrm{100sin}\:^{\mathrm{2}} \mathrm{t}\left(\mathrm{100}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{t}\right.}\right)}\:= \\ $$$$\left.\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{2}\:\mathrm{dt}\:=\:\mathrm{2t}\:\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\:\mathrm{2}×\frac{\pi}{\mathrm{2}}=\pi\: \\ $$
Answered by Dwaipayan Shikari last updated on 03/Jan/21
$$\int_{\mathrm{0}} ^{\mathrm{100}} \frac{{dx}}{\:\sqrt{{x}}\sqrt{\mathrm{100}−{x}}}\:\:\:\:\:\:\:{x}=\mathrm{100}{u} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{du}}{\:\sqrt{{u}}\sqrt{\mathrm{1}−{u}}}=\beta\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\sqrt{\pi}.\sqrt{\pi}=\pi \\ $$