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I-sin-3-cosx-4-dx-

Question Number 122468 by SOMEDAVONG last updated on 17/Nov/20 $$\mathrm{I}.\int\mathrm{sin}\left(\mathrm{3}−\mathrm{cosx}\right)^{\mathrm{4}} \mathrm{dx}=? \\ $$ Answered by Dwaipayan Shikari last updated on 17/Nov/20 $$\int{sinx}\left(\mathrm{3}−{cosx}\right)^{\mathrm{4}} {dx}\:\:\:\:\mathrm{3}−{cosx}={t}\Rightarrow{sinx}=\frac{{dt}}{{dx}} \\ $$$$\int{t}^{\mathrm{4}}…

If-a-and-b-are-positive-integers-such-that-1-ab-divides-a-2-b-2-show-that-the-integer-a-2-b-2-1-ab-must-be-a-perfect-square-

Question Number 56912 by Kunal12588 last updated on 26/Mar/19 $$\mathrm{If}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\left(\mathrm{1}+{ab}\right)\:\mathrm{divides}\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{integer} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{1}+{ab}}\:\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$ Commented by kaivan.ahmadi last updated…

dxdy-1-x-2-y-2-

Question Number 122412 by mohammad17 last updated on 16/Nov/20 $$\int\int\frac{{dxdy}}{\mathrm{1}−{x}^{\mathrm{2}} {y}^{\mathrm{2}} } \\ $$ Answered by Olaf last updated on 16/Nov/20 $$\mathrm{I}\left({x},{y}\right)\:=\:\int\int\frac{{dxdy}}{\mathrm{1}−{x}^{\mathrm{2}} {y}^{\mathrm{2}} } \\…