Menu Close

Author: Tinku Tara

Question-227128

Question Number 227128 by Spillover last updated on 01/Jan/26 Answered by som(math1967) last updated on 01/Jan/26 $$\int_{\mathrm{0}\:\:} ^{\frac{\pi}{\mathrm{2}}} \frac{\sqrt{{tanx}}}{\mathrm{1}+\sqrt{{tanx}}}{dx} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\sqrt{{tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right)}{dx}}{\mathrm{1}+\sqrt{{tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right)}} \\ $$$${I}=\int_{\mathrm{0}}…

prove-i-1-n-i-sin-1-i-gt-1-n-n-1-

Question Number 227115 by MrAjder last updated on 01/Jan/26 $$\mathrm{prove}:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}{i}\:\mathrm{sin}\frac{\mathrm{1}}{{i}}>\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)} \\ $$ Answered by MrAjder last updated on 01/Jan/26 $$\forall{i}\in\mathbb{Z}^{+} ,\mathrm{0}<\frac{\mathrm{1}}{{i}}\leq\mathrm{1}\Rightarrow\mathrm{sin}\frac{\mathrm{1}}{{i}}>\frac{\mathrm{1}}{{i}}−\frac{\mathrm{1}}{\mathrm{6}{i}^{\mathrm{3}} }\Rightarrow{i}\:\mathrm{sin}\frac{\mathrm{1}}{{i}}>\mathrm{1}−\frac{\mathrm{1}}{\mathrm{6}{i}^{\mathrm{2}} }…

2025-2026-2027-2026-1-3-

Question Number 227124 by BaliramKumar last updated on 01/Jan/26 $$\sqrt[{\mathrm{3}}]{\mathrm{2025}×\mathrm{2026}×\mathrm{2027}\:+\:\mathrm{2026}}\:=\:?\:\:\:\:\:\:\: \\ $$ Answered by TonyCWX last updated on 01/Jan/26 $$\sqrt[{\mathrm{3}}]{\left({x}−\mathrm{1}\right)\left({x}\right)\left({x}+\mathrm{1}\right)+{x}} \\ $$$$=\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} −{x}+{x}} \\ $$$$=\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}}…