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Category: Mensuration

Question-216162

Question Number 216162 by Spillover last updated on 28/Jan/25 Answered by MrGaster last updated on 02/Feb/25 $$\mathrm{Let}\:{x}=\frac{\pi}{\mathrm{3}}−{t}\Rightarrow{dx}=−{dt} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \sqrt{\mathrm{sin}^{\mathrm{3}} {x}\mathrm{sin}\left(\frac{\pi}{\mathrm{3}}−{x}\right)}{dx}\int_{\frac{\pi}{\mathrm{3}}} ^{\mathrm{0}} \sqrt{\mathrm{sin}^{\mathrm{3}} \left(\frac{\pi}{\mathrm{3}}−{t}\right)\mathrm{sin}\:{t}}\left(−{dt}\right)=\int_{\mathrm{0}}…

Question-212239

Question Number 212239 by Spillover last updated on 07/Oct/24 Answered by Ghisom last updated on 08/Oct/24 $$\int\frac{\mathrm{e}^{−{x}/\mathrm{2}} \sqrt{\mathrm{1}−\mathrm{sin}\:{x}}}{\mathrm{1}+\mathrm{cos}\:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\right] \\ $$$$=\int\frac{\mathrm{e}^{−\mathrm{arctan}\:{t}} \left({t}−\mathrm{1}\right)}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}= \\…

Question-212209

Question Number 212209 by efronzo1 last updated on 06/Oct/24 $$\:\:\:\cancel{\underbrace{\gtrdot}}\: \\ $$ Answered by som(math1967) last updated on 06/Oct/24 $$\:{let}\:{no}\:{of}\:{solders}\:{in}\:{A},{B},{C}\:{are} \\ $$$${x},{y},{z} \\ $$$$\therefore\mathrm{37}{x}+\mathrm{23}{y}=\mathrm{29}{x}+\mathrm{29}{y} \\…