Category: Integration

Question Number 28613 by abdo imad last updated on 27/Jan/18 $$letgivex>0andS\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{sint}{e^{xt}-1}dt.$$$$developpSatformofseries.$$ Commented by abdo imad last updated on…

Question Number 159680 by cortano last updated on 20/Nov/21 $${\int }_0^{\frac{\pi }{6}}\frac{\sin x\sin (x+60°)\sin (x+120°)}{\cos 3x+\sin 3x}dx=?$$ 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 28610 by abdo imad last updated on 27/Jan/18 $$letgiveI\left(x\right)={\int }_0^{\frac{\pi }{2}}\frac{dt}{\sqrt{{sin}^{2}t+x^2{cos}^{2}t}}and$$$${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 94143 by john santu last updated on 17/May/20 $$\int \frac{dx}{{(x+\sqrt{1+x^2})}^2}=$$ 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 94123 by  M±th+et+s last updated on 17/May/20 $$provethat$$$${\int }_0^{\mathrm{\infty }}\frac{{sin}^{2n}\left(x\right)}{x^2}d={\int }_0^{\mathrm{\infty }}\frac{{sin}^{2n-1}\left(x\right)}{x}dx$$$$$$$$$$…