Category: Integration

Question Number 31507 by abdo imad last updated on 09/Mar/18 $${g}\:{is}\:{real}\:{function}\:{continue}\:{let} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{sin}\left({x}−{t}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}^{'} \left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {cos}\left({t}−{x}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{so}<{ution}\:{of}\:{the}\:{diff}.{equa}. \\ $$$${y}^{''} \:+{y}\:={g}\left({x}\right)…

Question Number 97041 by bemath last updated on 06/Jun/20 $$\int\:\mathrm{sin}\:^{\mathrm{8}} \left({x}\right)\:\mathrm{cos}\:^{\mathrm{8}} \left({x}\right)\:{dx}\:=\:? \\ $$ Answered by john santu last updated on 06/Jun/20 $$\Rightarrow\mathrm{sin}\:^{\mathrm{8}} {x}.\mathrm{cos}\:^{\mathrm{8}} {x}\:=\:\frac{\mathrm{sin}\:^{\mathrm{8}}…

Question Number 31503 by abdo imad last updated on 09/Mar/18 $${find}\:\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} \:\:\:\:\:\frac{{dt}}{{t}\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}\:. \\ $$ Commented by abdo imad last updated on 16/Mar/18 $${ch}.\:{t}={ch}\left({x}\right)\:{give}\:\:{I}\:=\:\int_{{argch}\left(\mathrm{2}\right)}…

Question Number 162575 by Ar Brandon last updated on 30/Dec/21 $$\int_{\mathrm{0}} ^{\pi} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{sin}{x}}{dx} \\ $$ Answered by phanphuoc last updated on 30/Dec/21 $${you}\:{can}\:{put}\:{x}={pi}−{t} \\…

Question Number 31466 by abdo imad last updated on 08/Mar/18 $${let}\:{give}\:{I}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{I}_{{n}} \:\:\:. \\ $$ Terms of…