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Category: Integration

find-dx-x-1-x-2-

Question Number 31516 by abdo imad last updated on 09/Mar/18 $${find}\:\int\:\:\:\frac{{dx}}{{x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:. \\ $$ Commented by abdo imad last updated on 16/Mar/18 $${I}\:=\:\int\:\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:−{x}\right){dx}=\:\int\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\:{dx}\:−\frac{{x}^{\mathrm{2}}…

g-is-real-function-continue-let-f-x-0-x-sin-x-t-g-t-dt-1-prove-that-f-x-0-x-cos-t-x-g-t-dt-2-prove-that-f-is-so-lt-ution-of-the-diff-equa-y-y-g-x-

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)…

sin-8-x-cos-8-x-dx-

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}}…