Question Number 122828 by bemath last updated on 20/Nov/20 Answered by bobhans last updated on 20/Nov/20 $$\:{solve}\:\underset{\pi/\mathrm{4}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\:.\: \\ $$$$\:{Solution}\::\: \\ $$$${B}\left({x}\right)=\:\int\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}}\:{dx}\: \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}}\:\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:{x}}}…
Question Number 122823 by bemath last updated on 19/Nov/20 Answered by liberty last updated on 19/Nov/20 $$\:{L}\left({x}\right)\:=\:\int\:\frac{\mathrm{1}}{\:\sqrt{{x}\left(\mathrm{1}+\sqrt{{x}}\right)}}\:{dx}\:=\:\int\:\frac{\mathrm{1}}{\:\sqrt{{x}}\:\sqrt{\mathrm{1}+\sqrt{{x}}}\:}\:{dx} \\ $$$${let}\:{u}\:=\:\mathrm{1}+\sqrt{{x}}\:\Rightarrow{du}\:=\:\frac{{dx}}{\mathrm{2}\sqrt{{x}}} \\ $$$${L}\left({x}\right)\:=\:\mathrm{2}\int\:\frac{\mathrm{1}}{\:\sqrt{{u}}}\:{du}\:=\:\mathrm{4}\sqrt{{u}}\:+\:{c}\:\: \\ $$$${L}\left({x}\right)=\:\mathrm{4}\sqrt{\mathrm{1}+\sqrt{{x}}}\:+\:{c}\: \\ $$…
Question Number 122822 by bemath last updated on 19/Nov/20 Answered by liberty last updated on 20/Nov/20 $${I}=\int\:{x}\:\mathrm{tan}\:{x}\:\mathrm{sec}\:^{\mathrm{2}} {x}\:{dx}\:=\:\int\:{x}\:\mathrm{tan}\:{x}\:{d}\left(\mathrm{tan}\:{x}\right) \\ $$$${by}\:{D}.{I}\:{method}\:\rightarrow\begin{cases}{{u}={x}\rightarrow{du}={dx}}\\{{v}=\int\mathrm{tan}\:{x}\:{d}\left(\mathrm{tan}\:{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\:^{\mathrm{2}} {x}}\end{cases} \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}}{x}.\mathrm{tan}\:^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{tan}\:^{\mathrm{2}} {x}\:{dx}\:…
Question Number 122807 by TITA last updated on 19/Nov/20 Commented by TITA last updated on 19/Nov/20 $${please}\:{help} \\ $$ Answered by ebi last updated on…
Question Number 57241 by Aditya789 last updated on 01/Apr/19 $$\int\frac{×\sqrt{\mathrm{x}+\mathrm{1}}}{\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19 $${x}+\mathrm{1}={t}^{\mathrm{2}} \rightarrow{dx}=\mathrm{2}{tdt} \\ $$$$\int\frac{\left({t}^{\mathrm{2}} −\mathrm{1}\right){t}×\mathrm{2}{tdt}}{{t}^{\mathrm{2}} +\mathrm{1}}…
Question Number 57237 by maxmathsup by imad last updated on 31/Mar/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{sin}\left({xt}^{\mathrm{2}} −\mathrm{1}\right)}{{t}^{\mathrm{4}} \:+\mathrm{1}}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{2}} \:{cos}\left({xt}^{\mathrm{2}} −\mathrm{1}\right)}{{t}^{\mathrm{4}} \:+\mathrm{1}}\:{dt}…
Question Number 57236 by maxmathsup by imad last updated on 31/Mar/19 $${clalculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{t}^{\mathrm{2}{n}} \left(\mathrm{1}−{t}\right)^{{n}} {dt}\:\:\:{with}\:{n}\:{integr}\:{natural}\:. \\ $$ Commented by maxmathsup by imad last…
Question Number 57235 by maxmathsup by imad last updated on 31/Mar/19 $${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{cosx}\:−{sinx}}{\:\sqrt{{cos}^{\mathrm{8}} {x}\:+{sin}^{\mathrm{8}} {x}}}\:{dx} \\ $$ Commented by maxmathsup by imad last updated…
Question Number 57233 by maxmathsup by imad last updated on 31/Mar/19 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{x}^{\mathrm{4}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$ Commented by maxmathsup by imad…
Question Number 57231 by maxmathsup by imad last updated on 31/Mar/19 $${find}\:{tbe}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:\:\frac{{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}\:+\mathrm{2}\right)^{\mathrm{2}} }\:{dx} \\ $$ Commented by maxmathsup by imad last…