Question Number 28613 by abdo imad last updated on 27/Jan/18 $${let}\:{give}\:{x}>\mathrm{0}\:\:{and}\:{S}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sint}}{{e}^{{xt}} −\mathrm{1}}{dt}\:. \\ $$$${developp}\:{S}\:{at}\:{form}\:{of}\:{series}. \\ $$ Commented by abdo imad last updated on…
Question Number 159681 by mnjuly1970 last updated on 20/Nov/21 $$ \\ $$$$\:\:\:\:{prove}\:{that}\:: \\ $$$$\mathrm{P}=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}\:\right)\:\overset{?} {=}\:\frac{−\sqrt{\mathrm{2}}\:{sin}\left(\pi\sqrt{\mathrm{2}}\:\right)}{\pi} \\ $$$$\:\:\:\:\:{m}.{n} \\ $$ Answered by mindispower last…
Question Number 159680 by cortano last updated on 20/Nov/21 $$\:\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{sin}\:{x}\:\mathrm{sin}\:\left({x}+\mathrm{60}°\right)\:\mathrm{sin}\:\left({x}+\mathrm{120}°\right)}{\mathrm{cos}\:\mathrm{3}{x}\:+\:\mathrm{sin}\:\mathrm{3}{x}}\:{dx}=? \\ $$ Commented by cortano last updated on 20/Nov/21 $$\:{let}\:\theta=\mathrm{3}{x}\:\Rightarrow{I}=\frac{\mathrm{1}}{\mathrm{12}}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{sin}\:\:\theta}{\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}\:{d}\theta \\…
Question Number 28611 by abdo imad last updated on 27/Jan/18 $$\left.{let}\:{give}\:\theta\in\right]\mathrm{0},\pi\left[\:\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{e}^{−{i}\theta} −{t}}=\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{{in}\theta} }{{n}}\:\:.\right. \\ $$ Commented by abdo imad last updated…
Question Number 28610 by abdo imad last updated on 27/Jan/18 $${let}\:{give}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\:\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} \:{cos}^{\mathrm{2}} {t}}}\:\:{and} \\ $$$${J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cost}}{\:\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} {cos}^{\mathrm{2}} {t}}}{dt}\:{cslculate}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left({I}\left({x}\right)−{J}\left({x}\right)\right)…
Question Number 159682 by cortano last updated on 20/Nov/21 $$\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}\:+\:\mathrm{sin}\:{x}}\:{dx}\:=?\: \\ $$ Answered by Ar Brandon last updated on 20/Nov/21 $${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{cos}{x}\mathrm{sin}{x}}{\mathrm{cos}{x}+\mathrm{sin}{x}}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}}…
Question Number 94143 by john santu last updated on 17/May/20 $$\int\:\frac{{dx}}{\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\mathrm{2}} }\:= \\ $$ Commented by abdomathmax last updated on 17/May/20 $${I}\:=\int\:\:\frac{{dx}}{\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\mathrm{2}} }\:{we}\:{do}\:{the}\:{chsngement}\:{x}\:={sh}\left({t}\right)…
Question Number 159671 by Sameza last updated on 19/Nov/21 $$\int\underset{\mathrm{0}} {\overset{\infty} {\:}}\:\frac{\mathrm{sin}^{\mathrm{2}} \left({x}\right)−{x}\mathrm{sin}\left({x}\right)}{{x}^{\mathrm{3}} }\:{dx} \\ $$ Answered by metamorfose last updated on 20/Nov/21 $${u}\:{may}\:{use}\:{a}\:{DL}\:{of}\:\:\frac{{sinx}−{x}}{{x}^{\mathrm{2}} }…
Question Number 94123 by M±th+et+s last updated on 17/May/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{sin}^{\mathrm{2}{n}} \left({x}\right)}{{x}^{\mathrm{2}} }{d}=\int_{\mathrm{0}} ^{\infty} \frac{{sin}^{\mathrm{2}{n}−\mathrm{1}} \left({x}\right)}{{x}}{dx} \\ $$$$ \\ $$$$ \\ $$…
Question Number 94119 by i jagooll last updated on 17/May/20 $$\int\:\mathrm{cot}^{−\mathrm{1}} \left(\sqrt{\mathrm{x}}\right)\:\mathrm{dx}\: \\ $$ Commented by i jagooll last updated on 17/May/20 $${u}\:=\:\mathrm{cot}^{−\mathrm{1}} \left(\sqrt{{x}}\right)\:\Rightarrow\:{du}\:=\:\frac{{dx}}{\mathrm{2}\sqrt{{x}}\:\left(\mathrm{1}−{x}\right)} \\…