Question Number 227055 by Spillover last updated on 28/Dec/25 $${A}\:{parabolic}\:{refector}\:{is}\:{formed}\:{by} \\ $$$${revolving}\:{the}\:{arc}\:{of}\:{the}\:{parabala} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{ax}\:\:{from}\:{x}=\mathrm{0}\:\:\:\:{to}\:\:{x}={h} \\ $$$${about}\:{the}\:{axis}.{If}\:{the}\:\:{diameter} \\ $$$${of}\:{the}\:{reflector}\:{is}\:\mathrm{2}{l}.{Show}\:{that} \\ $$$${the}\:{area}\:{of}\:{the}\:{reflecting}\:{surface}\:{is} \\ $$$$\frac{\pi{l}}{\mathrm{6}{h}^{\mathrm{2}} }\left\{\left({l}^{\mathrm{2}} +\mathrm{4}{h}^{\mathrm{2}}…
Question Number 226994 by Spillover last updated on 24/Dec/25 Answered by Spillover last updated on 25/Dec/25 Answered by Spillover last updated on 25/Dec/25 Answered by…
Question Number 226995 by Spillover last updated on 24/Dec/25 Answered by Ghisom_ last updated on 24/Dec/25 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{\mathrm{sin}^{\mathrm{3}} \:{x}\:\mathrm{cos}^{\mathrm{5}} \:{x}}{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}^{\mathrm{2}} \:{x}\right)^{\mathrm{3}} }{dx}=\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}\right)\mathrm{sin}^{\mathrm{3}}…
Question Number 226952 by Spillover last updated on 20/Dec/25 Answered by Spillover last updated on 24/Dec/25 $$\left({a}\right) \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{−{x}} \mathrm{cos}\:^{{n}} {xdx} \\…
Question Number 226953 by Spillover last updated on 20/Dec/25 $${If}\:{I}_{{n}} =\int\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{{n}} {dx}\: \\ $$$${Show}\:{that} \\ $$$${I}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}{x}\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{{n}} +\mathrm{2}{na}^{\mathrm{2}} {I}_{{n}−\mathrm{1}} \: \\…
Question Number 226818 by Spillover last updated on 16/Dec/25 $${Evaluate} \\ $$$$\int\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{1}}{\mathrm{2}{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}}{dx} \\ $$ Answered by Frix last updated on 16/Dec/25 $$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dx}}{{x}}−\frac{\mathrm{1}}{\mathrm{10}}\int\frac{{dx}}{{x}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{5}}\int\frac{{dx}}{\mathrm{2}{x}−\mathrm{1}}=…
Question Number 226819 by Spillover last updated on 16/Dec/25 $${Evaluate} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$ Answered by AgniMath last updated on 16/Dec/25 $$\int_{\mathrm{0}}…
Question Number 226799 by CrispyXYZ last updated on 15/Dec/25 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}\:\mathrm{d}{x}\:=\:? \\ $$ Answered by breniam last updated on 17/Dec/25 $$\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{y}\underset{\mathrm{0}}…
Question Number 226780 by Spillover last updated on 14/Dec/25 $${By}\:{using}\:{concept}\:{of}\:{complex} \\ $$$${number} \\ $$$${show}\:{that} \\ $$$$\mathrm{tan}\:\mathrm{5}\theta=\frac{\mathrm{tan}\:^{\mathrm{5}} \theta−\mathrm{10tan}\:^{\mathrm{3}} \theta+\mathrm{5tan}\:\theta}{\mathrm{5tan}\:^{\mathrm{4}} \theta−\mathrm{10tan}\:^{\mathrm{2}} \theta+\mathrm{1}} \\ $$ Answered by Frix…
Question Number 226776 by Spillover last updated on 14/Dec/25 $${Approximate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {xe}^{{x}^{\mathrm{2}} } {dx}\:{with}\:\mathrm{6}\:{ordinates}. \\ $$$${Use}\:{both}\:{rules}\:{Simpsons}\:{and} \\ $$$${Trapozoidal}\:{rules},{hence}\:{evaluate}\:{and} \\ $$$${calculate}\:{the}\:{percentage}\:{error} \\ $$$${commetted}\:{for}\:{each}\:{case}.{Give}\:{comments} \\ $$$$ \\…