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Category: Integration

sin-2-x-cos-4x-dx-

Question Number 131068 by EDWIN88 last updated on 01/Feb/21 $$\:\int\:\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:\mathrm{4}{x}\:{dx}\:=?\: \\ $$ Answered by Ar Brandon last updated on 01/Feb/21 $$\mathcal{I}=\int\mathrm{sin}^{\mathrm{2}} \mathrm{xcos4xdx}=\frac{\mathrm{1}}{\mathrm{2}}\int\left(\mathrm{1}+\mathrm{cos2x}\right)\mathrm{cos4xdx} \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int\left[\mathrm{cos4x}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos6x}+\mathrm{cos2x}\right)\right]\mathrm{dx}…

x-2-2-x-2-3-x-2-4-x-2-5-x-2-6-x-2-7-dx-

Question Number 131058 by pipin last updated on 01/Feb/21 $$\int\frac{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{5}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{6}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{7}\right)}\:\mathrm{dx}\: \\ $$ Answered by MJS_new last updated on 01/Feb/21…

x-2-2-x-4-4-dx-

Question Number 131053 by pipin last updated on 01/Feb/21 $$\int\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}^{\mathrm{4}} +\mathrm{4}}\mathrm{dx} \\ $$ Answered by Ar Brandon last updated on 01/Feb/21 $$\mathcal{I}=\int\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}^{\mathrm{4}} +\mathrm{4}}\mathrm{dx}=\int\frac{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{2}}…

let-f-x-0-tsin-xt-x-2-t-2-2-dt-x-gt-0-calculate-f-x-

Question Number 131050 by mathmax by abdo last updated on 31/Jan/21 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{tsin}\left(\mathrm{xt}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dt}\:\:\:\left(\mathrm{x}>\mathrm{0}\right) \\ $$$$\mathrm{calculate}\:\mathrm{f}^{'} \left(\mathrm{x}\right) \\ $$ Answered by mathmax…

let-f-x-0-cos-xt-x-2-t-2-dt-calculate-0-1-f-x-dx-

Question Number 131049 by mathmax by abdo last updated on 31/Jan/21 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{xt}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{t}^{\mathrm{2}} }\mathrm{dt}\:\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$ Answered by mindispower last updated…

Question-65486

Question Number 65486 by aliesam last updated on 30/Jul/19 Answered by MJS last updated on 31/Jul/19 $$\int\mathrm{arctan}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\:{dx}= \\ $$$$\:\:\:\:\:{u}'=\mathrm{1}\:\rightarrow\:{u}={x} \\ $$$$\:\:\:\:\:{v}=\mathrm{arctan}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\:\rightarrow\:{v}'=−\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right)}…