Category: Integration

Question Number 63667 by mathmax by abdo last updated on 07/Jul/19 $$1)calculate{\int }_0^{2\pi }\frac{dt}{cost+xsint}wihxfromR.$$$$2)calculate{\int }_0^{2\pi }\frac{sint}{{(cost+xsint)}^2}dt$$$$3)find[thevalueof{\int }_0^{2\pi }\frac{dt}{cos\left(2t\right)+2sin\left(2t\right)}$$…

Question Number 63664 by mathmax by abdo last updated on 07/Jul/19 $$letf\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{t^{a-1}}{x+t^n}dtwith0<a<1andx>0andn⩾2$$$$1)determineaexplicitformoff\left(x\right)$$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}^{{n}} \right)^{\mathrm{2}} }\:{dt}…

Question Number 63662 by mathmax by abdo last updated on 06/Jul/19 $$let{A}_n={\int }_0^{\mathrm{\infty }}\frac{x^{a-1}}{1+x^n}dxwithnintegrandn⩾2and0<a<1$$$$1)calculate{A}_n$$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{and}\:\int_{\mathrm{0}}…

Question Number 63641 by aliesam last updated on 06/Jul/19 Commented by mathmax by abdo last updated on 06/Jul/19 $${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mid{sin}\left({n}\pi{x}\right)\mid}{{x}^{\mathrm{2}} \:+\mathrm{1}}\:{dx}\:\Rightarrow\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ={lim}_{{n}\rightarrow+\infty}…

Question Number 129173 by bramlexs22 last updated on 13/Jan/21 $$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\tan \theta +\cot \theta +\sec \theta +\csc \theta }?$$ Answered by liberty last updated on 13/Jan/21 $$\mathrm{J}=\int \frac{\mathrm{d}\theta }{\frac{\sin \theta +1}{\cos \theta }+\frac{\cos \theta +1}{\sin \theta }}=\int \frac{\cos \theta \sin \theta \mathrm{d}\theta }{\sin {}^2\theta +\sin \theta +\cos {}^2\theta +\cos \theta }$$$$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{d}\theta}{\mathrm{1}+\mathrm{sin}\:\theta+\mathrm{cos}\:\theta}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\theta}{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}\mathrm{d}\theta…