Question Number 77242 by Tony Lin last updated on 04/Jan/20 $${prove}\:{that} \\ $$$$\:\int_{\mathrm{0}} ^{{a}} \sqrt{\mathrm{2}+\frac{{a}}{{x}}−\mathrm{2}\sqrt{\frac{{a}}{{x}}}\:}{dx}={a}\left[\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{ln}\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)+\mathrm{1}\right] \\ $$ Commented by mathmax by abdo last updated on…
Question Number 142763 by lapache last updated on 05/Jun/21 Commented by Ar Brandon last updated on 05/Jun/21 $$\mathcal{I}=\int\sqrt{\mathrm{tanx}}\mathrm{dx}\:,\:\mathrm{t}^{\mathrm{2}} =\mathrm{tanx}\:\Rightarrow\mathrm{2tdt}=\left(\mathrm{1}+\mathrm{t}^{\mathrm{4}} \right)\mathrm{dx} \\ $$$$\:\:\:=\int\frac{\mathrm{2t}^{\mathrm{2}} \mathrm{dt}}{\mathrm{1}+\mathrm{t}^{\mathrm{4}} }=\int\frac{\left(\mathrm{t}^{\mathrm{2}} +\mathrm{1}\right)+\left(\mathrm{t}^{\mathrm{2}}…
Question Number 77221 by john santu last updated on 04/Jan/20 $$\int\:\frac{\sqrt{\mathrm{2}+{x}}\:{dx}}{\:\sqrt{{x}^{\mathrm{5}} }}\:=\:? \\ $$ Answered by jagoll last updated on 04/Jan/20 $$\int\:\frac{\sqrt{\mathrm{2}+\mathrm{x}}}{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}}}\mathrm{dx}\:=\:\int\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\sqrt{\frac{\mathrm{2}}{\mathrm{x}}+\mathrm{1}}\:\mathrm{dx} \\…
Question Number 11660 by Nayon last updated on 29/Mar/17 $$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:{Evaluate}\:\:\int{x}^{{x}} {dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$…
Question Number 142719 by gsk2684 last updated on 04/Jun/21 $$\mathrm{find}\:\mathrm{the}\:\mathrm{particular}\:\mathrm{solution} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\mathrm{y}^{\left(\mathrm{4}\right)} +\mathrm{21y}^{\left(\mathrm{2}\right)} −\mathrm{100y}=\mathrm{4}\left(\mathrm{8}−\mathrm{29t}\right)\mathrm{e}^{−\mathrm{2t}} . \\ $$$$\mathrm{solution}\:\mathrm{please}. \\ $$ Commented by gsk2684 last…
Question Number 142708 by ajfour last updated on 05/Jun/21 $$\:{I}=\int_{\mathrm{0}} ^{\:\:\alpha} \sqrt{{c}^{\mathrm{2}} −\mathrm{sin}\:^{\mathrm{2}} \theta}{d}\theta \\ $$$$\:\mathrm{tan}\:\alpha=\frac{{a}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:\:,\:{a}^{\mathrm{2}} >{b}^{\mathrm{2}} \:\:,\:{c}^{\mathrm{2}} >\mathrm{1} \\ $$$${Perimeter}\:{of}\:{ellipse} \\ $$$$=\mathrm{4}\int_{\mathrm{0}}…
Question Number 11621 by Nayon last updated on 29/Mar/17 $${Evaluate}\:\:\int\frac{{ax}+{b}}{{cx}+{d}}{dx} \\ $$ Answered by mrW1 last updated on 29/Mar/17 $$=\int\frac{\frac{{a}}{{c}}\left({cx}+{d}\right)+\left({b}−\frac{{ad}}{{c}}\right)}{{cx}+{d}}{dx} \\ $$$$=\frac{{a}}{{c}}\int{dx}+\left({b}−\frac{{ad}}{{c}}\right)×\frac{\mathrm{1}}{{c}}\int\frac{{d}\left({cx}+{d}\right)}{{cx}+{d}} \\ $$$$=\frac{{ax}}{{c}}+\frac{{bc}−{ad}}{{c}^{\mathrm{2}} }×\mathrm{ln}\:\left({cx}+{d}\right)+{C}…
Question Number 77158 by TawaTawa last updated on 03/Jan/20 $$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{3}} \:\mathrm{cos}\frac{\mathrm{x}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}\:−\:\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$ Commented by TawaTawa last updated on 03/Jan/20 $$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\…
Question Number 142689 by rs4089 last updated on 04/Jun/21 Answered by Dwaipayan Shikari last updated on 04/Jun/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}\right)}{{x}}{f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({x}\right){dx}\:\:\:\:{f}\left({x}\pm\pi\right)={f}\left({x}\right) \\ $$$${let}\:{f}\left({x}\right)={sin}^{\mathrm{2}{n}−\mathrm{2}} {x}\:\:\:\:\:\:…
Question Number 11619 by Nayon last updated on 29/Mar/17 $${Evaluate}\:\int\frac{{dx}}{\left({x}+{a}\right)\left({x}+{b}\right)} \\ $$ Commented by Nayon last updated on 29/Mar/17 $${someone}\:{pls}\:{ans} \\ $$ Answered by mrW1…