Question Number 58488 by Mr X pcx last updated on 23/Apr/19 $${let}\:{f}\left({x}\right)\:=\int\:\:\:\frac{{dt}}{{x}\:+{cost}\:+{cos}\left(\mathrm{2}{t}\right)}\:\:\left({x}\:{real}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){determine}\:{also}\:\int\:\:\frac{{dt}}{\left({x}+{cost}\:+{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:\:\:\frac{{dt}}{\mathrm{1}+{cos}\left({t}\right)+{cos}\left(\mathrm{2}{t}\right)}\:{and} \\ $$$$\int\:\:\:\frac{{dt}}{\left(\mathrm{3}\:+{cos}\left({t}\right)+{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} } \\ $$ Commented…
Question Number 124024 by hatakekakashi1729gmailcom last updated on 30/Nov/20 Answered by Dwaipayan Shikari last updated on 30/Nov/20 $$\left({x}^{{x}} \right)^{\left({x}^{{x}} \right)^{\left({x}^{\left.{x}\right)} \right)…} } ={t}\:\:\:\:\:\:\:\:\:\:\:{x}^{{x}} ={y} \\…
Question Number 58487 by Mr X pcx last updated on 23/Apr/19 $${let}\:{f}\left({x}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dt}}{\mathrm{2}+{xsint}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){determine}\:{also}\:{g}\left({x}\right)=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{sint}}{\left(\mathrm{2}+{xsint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\frac{{dt}}{\mathrm{2}+\mathrm{3}{sint}}…
Question Number 189549 by mnjuly1970 last updated on 18/Mar/23 $$ \\ $$$$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{dxdy}}{\left(\mathrm{1}+{xy}\:\right)^{\:\mathrm{4}} }=? \\ $$ Answered by witcher3 last updated on…
Question Number 58478 by rahul 19 last updated on 23/Apr/19 $$\left\{\mathrm{1}\right\}\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{4}} +\mathrm{8}{x}^{\mathrm{2}} +\mathrm{4}}\:{dx}\:=\:? \\ $$$$\left\{\mathrm{2}\right\}\:\:{Shortest}\:{distance}\:{between}\:{the} \\ $$$${parabolas}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:{and}\:{y}^{\mathrm{2}} =\mathrm{2}{x}−\mathrm{6}\:{is}\:? \\ $$ Commented by maxmathsup…
Question Number 124004 by john_santu last updated on 30/Nov/20 $$\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}}}\:?\: \\ $$ Answered by liberty last updated on 30/Nov/20 $${T}\:=\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}}}\:;\:\left[\:{let}\:{u}^{\mathrm{3}} =\mathrm{tan}\:^{\mathrm{2}} {x}\:\wedge\:{dx}\:=\frac{\mathrm{3}{u}^{\mathrm{2}} }{\mathrm{2}{u}^{\mathrm{3}/\mathrm{2}} \:\left({u}^{\mathrm{3}} +\mathrm{1}\right)}{du}\:\right]…
Question Number 123998 by john_santu last updated on 30/Nov/20 $$\:\int_{\:\mathrm{0}} ^{\:\infty} \:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\: \\ $$ Answered by liberty last updated on 30/Nov/20 $$\:{substituting}\:{x}\:=\:\mathrm{tan}\:{q}\:{with}\:{upper}\:{limit}\:\frac{\pi}{\mathrm{2}}…
Question Number 123977 by mnjuly1970 last updated on 29/Nov/20 Answered by mathmax by abdo last updated on 29/Nov/20 $$\mathrm{let}\:\mathrm{A}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cosx}\right)\mathrm{dx}\:\:\mathrm{we}\:\mathrm{have}\:\mathrm{A}\:=−\frac{\pi}{\mathrm{2}}\mathrm{ln}\left(\mathrm{2}\right)\:\mathrm{also} \\ $$$$\mathrm{A}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\frac{\mathrm{e}^{\mathrm{ix}}…
Question Number 123967 by Algoritm last updated on 29/Nov/20 Answered by TANMAY PANACEA last updated on 29/Nov/20 $${t}^{\mathrm{6}} ={x}+\mathrm{2} \\ $$$$\int\frac{{t}}{{t}^{\mathrm{2}} +{t}^{\mathrm{3}} }×\mathrm{6}{t}^{\mathrm{5}} {dt} \\…
Question Number 123964 by mindispower last updated on 29/Nov/20 $$\int\sqrt{{x}^{\mathrm{3}} +{ax}+{b}}{dx} \\ $$ Commented by MJS_new last updated on 29/Nov/20 $${x}_{\mathrm{1}} ={p} \\ $$$${x}_{\mathrm{2}} =−\frac{{p}}{\mathrm{2}}−\sqrt{{q}}…