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Author: Tinku Tara

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Question Number 208238 by alcohol last updated on 08/Jun/24 $$\mathrm{S}{how}\:{that} \\ $$$$\frac{\pi}{\mathrm{4}}\:<\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:{using}\:{x}\:=\:{sint} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}<\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$$${using}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){g}\left({x}\right){dx}\right)^{\mathrm{2}} <\int_{\mathrm{0}}…

Question-208235

Question Number 208235 by efronzo1 last updated on 08/Jun/24 Answered by som(math1967) last updated on 08/Jun/24 $$\:{here}\:{f}\left({x}\right)={f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\:\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}{dx} \\ $$$$=\underset{\mathrm{2}} {\overset{\mathrm{4}}…