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Category: Integration

math-analysis-xsin-x-x-2-2x-2-dx-xsin-x-x-1-2-1-dx-x-1-t-t-1-sin-t-1-t-2-1-dt-

Question Number 131969 by mnjuly1970 last updated on 10/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:{math}\:\:{analysis}… \\ $$$$\:\:\:\:\phi=\:\:\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}{dx}=? \\ $$$$\:\:\:\:\:\phi=\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$$$\:\:\:\:\:\:\:\overset{{x}+\mathrm{1}={t}} {=}\int_{−\infty} ^{\:+\infty} \frac{\left({t}−\mathrm{1}\right){sin}\left({t}−\mathrm{1}\right)}{{t}^{\mathrm{2}}…

sin-4-x-cos-4-x-dx-

Question Number 131961 by Zack_ last updated on 10/Feb/21 $$\int\left({sin}^{\mathrm{4}} {x}.{cos}^{\mathrm{4}} {x}\right){dx} \\ $$ Answered by liberty last updated on 10/Feb/21 $$\mathrm{I}=\int\:\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{2}} \mathrm{dx} \\…

Question-131919

Question Number 131919 by Algoritm last updated on 09/Feb/21 Answered by Dwaipayan Shikari last updated on 09/Feb/21 $${x}={u}^{\mathrm{6}} \\ $$$$=\mathrm{6}\int\frac{{u}^{\mathrm{5}} }{{u}^{\mathrm{2}} +{u}^{\mathrm{3}} }{du}=\mathrm{6}\int\frac{{u}^{\mathrm{3}} }{{u}+\mathrm{1}}{du}=\mathrm{6}\int\left({u}^{\mathrm{2}} −{u}+\mathrm{1}\right){du}−\mathrm{6}{log}\left({u}\right)…