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Category: Integration

Question-84693

Question Number 84693 by Power last updated on 15/Mar/20 Commented by abdomathmax last updated on 15/Mar/20 $${let}\:\varphi\left({t}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({tx}\right)}{{x}\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)}{dx}\:\:{we}\:{have} \\ $$$$\varphi^{'} \left({t}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({tx}\right)}{{x}^{\mathrm{2}}…

Question-84681

Question Number 84681 by Power last updated on 15/Mar/20 Commented by mathmax by abdo last updated on 15/Mar/20 $${I}\:=\int\:\:\frac{\mathrm{15}{sinx}\:+\mathrm{2}{cosx}}{\mathrm{98}{sinx}\:−\mathrm{7}{cosx}}{dx}\:\Rightarrow{I}\:=\frac{\mathrm{15}}{\mathrm{98}}\int\:\:\frac{{sinx}+\frac{\mathrm{2}}{\mathrm{15}}{cosx}}{{sinx}−\frac{\mathrm{7}}{\mathrm{98}}{cosx}}{dx} \\ $$$${let}\:{determine}\:{A}\:=\int\:\:\frac{{sinx}+{acosx}}{{sinx}\:+{bcosx}}{dx}\:\:{we}\:{di}\:{the}\:{changement} \\ $$$${tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:\Rightarrow\:{A}\:=\int\frac{\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }+{a}\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}}…

x-sin-1-x-dx-

Question Number 84666 by jagoll last updated on 14/Mar/20 $$\int\:{x}\:\mathrm{sin}^{−\mathrm{1}} \left({x}\right)\:{dx}\: \\ $$ Commented by mathmax by abdo last updated on 15/Mar/20 $${A}\:=\int\:{xarcsinx}\:{dx}\:\:{by}\:{parts}\:\:{u}^{'} ={x}\:{and}\:{v}={arcsinx}\:\Rightarrow \\…

sec-3-x-dx-

Question Number 19077 by Joel577 last updated on 04/Aug/17 $$\int\:\mathrm{sec}^{\mathrm{3}} \:{x}\:{dx} \\ $$ Answered by ajfour last updated on 04/Aug/17 $$\mathrm{I}=\int\mathrm{sec}\:\theta\left(\mathrm{sec}\:^{\mathrm{2}} \theta\mathrm{d}\theta\right) \\ $$$$=\mathrm{sec}\:\theta\int\mathrm{sec}\:^{\mathrm{2}} \theta\mathrm{d}\theta−\int\left[\frac{\mathrm{d}}{\mathrm{d}\theta}\left(\mathrm{sec}\:\theta\right).\int\mathrm{sec}\:^{\mathrm{2}}…

calculate-dx-cosx-cos-2x-cos-3x-

Question Number 84577 by msup trace by abdo last updated on 14/Mar/20 $${calculate}\:\int\:\:\:\:\frac{{dx}}{{cosx}\:+{cos}\left(\mathrm{2}{x}\right)+{cos}\left(\mathrm{3}{x}\right)} \\ $$ Commented by jagoll last updated on 14/Mar/20 $$\mathrm{cos}\:{x}+\mathrm{cos}\:\mathrm{3}{x}+\mathrm{cos}\:\mathrm{2}{x}\:=\: \\ $$$$\mathrm{2cos}\:\mathrm{2}{x}\:\mathrm{cos}\:{x}\:+\:\mathrm{cos}\:\mathrm{2}{x}\:=\:…