Question Number 63641 by aliesam last updated on 06/Jul/19 Commented by mathmax by abdo last updated on 06/Jul/19 $${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mid{sin}\left({n}\pi{x}\right)\mid}{{x}^{\mathrm{2}} \:+\mathrm{1}}\:{dx}\:\Rightarrow\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ={lim}_{{n}\rightarrow+\infty}…
Question Number 129177 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{sin}\:\mathrm{x}\left(\mathrm{sin}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx} \\ $$$$\:\mathrm{G}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}^{\mathrm{2}} }{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left\{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right\}}\mathrm{dx}…
Question Number 129173 by bramlexs22 last updated on 13/Jan/21 $$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{d}\theta}{\mathrm{tan}\:\theta+\mathrm{cot}\:\theta+\mathrm{sec}\:\theta+\mathrm{csc}\:\theta}\:? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\mathrm{J}\:=\:\int\frac{\mathrm{d}\theta}{\frac{\mathrm{sin}\:\theta+\mathrm{1}}{\mathrm{cos}\:\theta}+\frac{\mathrm{cos}\:\theta+\mathrm{1}}{\mathrm{sin}\:\theta}}\:=\:\int\:\frac{\mathrm{cos}\:\theta\:\mathrm{sin}\:\theta\:\mathrm{d}\theta}{\mathrm{sin}\:^{\mathrm{2}} \theta+\mathrm{sin}\:\theta+\mathrm{cos}\:^{\mathrm{2}} \theta+\mathrm{cos}\:\theta} \\ $$$$\:\mathrm{J}\:=\:\int\:\frac{\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta\:\mathrm{d}\theta}{\mathrm{1}+\mathrm{sin}\:\theta+\mathrm{cos}\:\theta}\:=\:\int\:\frac{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\theta}{\mathrm{2sin}\:\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\theta}{\mathrm{2}}\right)+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}\mathrm{d}\theta…
Question Number 129158 by gowsalya last updated on 13/Jan/21 Answered by MJS_new last updated on 13/Jan/21 $$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\mid\mathrm{cos}\:\theta\:+\mathrm{i}\:\mathrm{sin}\:\theta\:−\mathrm{1}\mid{d}\theta= \\ $$$$=\mathrm{2}\sqrt{\mathrm{2}}\underset{\mathrm{0}} {\overset{\pi} {\int}}\sqrt{\mathrm{1}−\mathrm{cos}\:\theta}\:{d}\theta= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\:\rightarrow\:{d}\theta=\frac{\mathrm{2}{dt}}{{t}^{\mathrm{2}}…
Question Number 129159 by gowsalya last updated on 13/Jan/21 Answered by TheSupreme last updated on 13/Jan/21 $$\int\left({e}^{{i}\theta} +{e}^{−{i}\theta} \right){e}^{{i}\theta} /\mathrm{2}=\frac{{e}^{{i}\theta} }{\mathrm{4}{i}}+\frac{\theta}{\mathrm{2}}+{c} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} …=\pi…
Question Number 63615 by aliesam last updated on 06/Jul/19 Commented by mathmax by abdo last updated on 06/Jul/19 $${changement}\:\:{x}=−{t}\:\:{give}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:{xln}\left(\mathrm{1}+\mathrm{10}^{{x}} \right){dx}\:=−\int_{−\mathrm{1}} ^{\mathrm{1}} \:\left(−{t}\right){ln}\left(\mathrm{1}+\mathrm{10}^{−{t}} \right)\left(−{dt}\right)…
Question Number 129150 by benjo_mathlover last updated on 13/Jan/21 $$\:\int_{\mathrm{0}} ^{\:\mathrm{x}} \left(\mathrm{x}−\mathrm{u}\right)^{\mathrm{2}} \mathrm{u}\:\mathrm{f}\left(\mathrm{u}\right)\:\mathrm{du}\:=\:\mathrm{4x}−\mathrm{6sin}\:\mathrm{x}+\mathrm{2xcos}\:\mathrm{x} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{0}}…
Question Number 63570 by mr W last updated on 05/Jul/19 Commented by mathmax by abdo last updated on 05/Jul/19 $$\int_{\mathrm{0}} ^{\mathrm{1}} \left(−\mathrm{1}\right)^{{x}} {dx}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{i}\pi{x}}…
Question Number 63566 by aliesam last updated on 05/Jul/19 $${prove}\:{that} \\ $$$$ \\ $$$$\int{sin}^{{n}} \left({x}\right)\:{dx}\:,\:{p}\in{n}\:,\:{p}\geqslant\mathrm{2}\:=−\:\frac{\mathrm{1}}{{n}}{cos}\left({x}\right)\:{sin}^{{n}−\mathrm{1}} \left({x}\right)\:+\:\left({p}−\mathrm{1}\right)\int{sin}^{{n}−\mathrm{2}} \left({x}\right)\:{dx} \\ $$ Commented by aliesam last updated on…
Question Number 129068 by bagjagunawan last updated on 12/Jan/21 Answered by mnjuly1970 last updated on 12/Jan/21 $$\:\:{ans}\:::\:\:\pi{ln}\left(\mathrm{1}−{e}^{−\pi{n}} \right)\:…. \\ $$ Terms of Service Privacy Policy…