Question Number 94135 by redmiiuser last updated on 17/May/20
$${f}\left({x}\right)\:{is}\:{a}\:{continuous}\:{function}\:{forall}\:{real}\:{values}\:{of}\:{x}\:{and}\:{satisfies}\int_{\mathrm{0}} ^{{x}} {f}\left({t}\right).{dt}=\int_{{x}} ^{\mathrm{1}} {t}^{\mathrm{2}} .{f}\left({t}\right).{dt}+\frac{{x}^{\mathrm{16}} }{\mathrm{8}}+\frac{{x}^{\mathrm{6}} }{\mathrm{3}}+{A} \\ $$$${Find}\:{A}? \\ $$
Commented by john santu last updated on 17/May/20
$$\mathrm{f}\left(\mathrm{x}\right)\:=\:−\mathrm{x}^{\mathrm{2}} \:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{2x}^{\mathrm{15}} +\mathrm{2x}^{\mathrm{5}} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{2x}^{\mathrm{15}} +\mathrm{2x}^{\mathrm{5}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }=\:\frac{\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{7}} \mathrm{x}+\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} \mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$=\:\mathrm{g}\left(\mathrm{x}\right)−\frac{\mathrm{4x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\: \\ $$$$\mathrm{A}\:=\:−\mathrm{4}\: \\ $$
Commented by redmiiuser last updated on 19/May/20
$${no}. \\ $$$${Wrong}\:{Answer}?? \\ $$