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Category: Integration

let-f-x-e-2x-actan-3x-1-1-calculste-f-n-x-and-f-n-0-2-if-f-x-a-n-x-n-determine-the-sequence-a-n-3-calculate-0-f-x-dx-

Question Number 126179 by mathmax by abdo last updated on 17/Dec/20 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{e}^{−\mathrm{2x}} \:\mathrm{actan}\:\left(\mathrm{3x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculste}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\Sigma\:\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \:\mathrm{determine}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{a}_{\mathrm{n}} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}…

Question-60637

Question Number 60637 by rajesh4661kumar@gamil.com last updated on 23/May/19 Commented by maxmathsup by imad last updated on 23/May/19 $$\int\:\:\:\:\frac{{sinx}}{\mathrm{1}−{sinx}}\:{dx}\:=−\int\frac{\mathrm{1}−{sinx}\:−\mathrm{1}}{\mathrm{1}−{sinx}}{dx}\:=−{x}\:+\int\:\:\frac{{dx}}{\mathrm{1}−{sinx}} \\ $$$$\int\:\:\frac{{dx}}{\mathrm{1}−{sinx}}\:=_{{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}} \:\:\:\:\:\:\int\:\:\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\mathrm{2}\:\int\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} −\mathrm{2}{t}}\:=\mathrm{2}\:\int\:\:\frac{{dt}}{\left({t}−\mathrm{1}\right)^{\mathrm{2}}…