Question Number 122751 by bemath last updated on 19/Nov/20 $$\:\underset{\mathrm{0}} {\overset{\mathrm{1}/\sqrt{\mathrm{2}}} {\int}}\:\frac{{x}\:\mathrm{sin}^{−\mathrm{1}} \left({x}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:{dx}\:?\: \\ $$ Answered by liberty last updated on 19/Nov/20 $$\:\:{Let}\:\mathrm{sin}^{−\mathrm{1}}…
Question Number 122748 by bemath last updated on 19/Nov/20 $$\:\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\frac{{dx}}{\left(\mathrm{3}−{x}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\:? \\ $$ Answered by MJS_new last updated on 19/Nov/20 $$\int\frac{{dx}}{\left(\mathrm{3}−{x}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}= \\…
Question Number 122734 by bemath last updated on 19/Nov/20 Answered by liberty last updated on 19/Nov/20 $$\:{Just}\:{applying}\:{partial}\:{faction} \\ $$$$\:\frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{6}{x}−\mathrm{4}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{2}\right)}\:=\:\frac{{Ax}+{B}}{{x}^{\mathrm{2}} +\mathrm{1}}+\:\frac{{Cx}+{D}}{{x}^{\mathrm{2}} +\mathrm{2}}…
Question Number 57194 by maxmathsup by imad last updated on 31/Mar/19 $${let}\:\:{A}_{{n}} =\int_{{n}} ^{{n}} \:\frac{\left[\sqrt{{x}+\mathrm{1}}\right]−\left[\sqrt{{x}}\right]}{{x}}\:{dx}\:\:\:{with}\:{n}\:{natural}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right){find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{A}_{{n}} \\ $$ Commented by maxmathsup…
Question Number 122713 by mnjuly1970 last updated on 19/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:\:{advanced}\:\:{math}\:… \\ $$$$\:\:\:\:\:{two}\:\:{simple}\:{and}\:{nice}\:{integrals}: \\ $$$$\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({e}^{−\gamma} {x}\right){ln}\left({x}\right)}{{x}}\:{dx}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Omega_{\mathrm{2}} \:=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}^{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}…
Question Number 122697 by john santu last updated on 19/Nov/20 $$\:\:\int\:\frac{{dx}}{\:\sqrt{\left({x}−{a}\right)\left({b}−{x}\right)}}\:? \\ $$ Answered by liberty last updated on 19/Nov/20 $$\:\:{Solve}\:\varphi\left({x}\right)=\int\:\frac{{dx}}{\:\sqrt{\left({x}−{a}\right)\left({b}−{x}\right)}}\:? \\ $$$$\:\:\:{Solution}\::\: \\ $$$$\:{letting}\:{x}\:=\:{a}\mathrm{cos}\:^{\mathrm{2}}…
Question Number 122689 by john santu last updated on 19/Nov/20 $$\:{Evaluate}\:{the}\:{integral}\: \\ $$$$\:\:\int\:\frac{\sqrt[{\mathrm{3}}]{{x}}\:+\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{{x}}\:−\mathrm{1}}\:{dx}\: \\ $$ Answered by bobhans last updated on 19/Nov/20 $$\:{let}\:{x}\:=\:{u}^{\mathrm{3}} \:\Rightarrow{dx}\:=\:\mathrm{3}{u}^{\mathrm{2}} \:{du}\:…
Question Number 122671 by mnjuly1970 last updated on 18/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{integral}… \\ $$$$\:\:\:{prove}\:{that}\:\::: \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {cos}\left(\pi{nx}\right)\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\frac{\pi{coth}\left(\pi{x}\right)}{{x}}\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{???} {=}\pi{ln}\left(\mathrm{1}−{e}^{−\pi{n}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}. \\ $$…
Question Number 188192 by universe last updated on 26/Feb/23 $$\:\:\: \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {e}^{−{a}^{\mathrm{2}} {x}^{\mathrm{2}} } \mathrm{cos}\left(\mathrm{2}{bx}\right)\:{dx}\:\:\:=\:\:\:\frac{\sqrt{\pi}}{\mathrm{2}{a}}{e}^{−{b}^{\mathrm{2}} /{a}^{\mathrm{2}} } \\ $$$$ \\ $$$$…
Question Number 122636 by mnjuly1970 last updated on 18/Nov/20 $$\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:{In}\:\:\mathrm{A}\overset{\Delta} {\mathrm{B}C}\:\:{prove}\:::\: \\ $$$$\:\:\:\ast:\:\:{sin}\left(\frac{\mathrm{A}}{\mathrm{2}}\right){sin}\left(\frac{\mathrm{B}}{\mathrm{2}}\right){sin}\left(\frac{\mathrm{C}}{\mathrm{2}}\right)\leqslant\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$……………………. \\ $$$$\:\:\:\:\ast\ast::\:\:\:{max}\left({cos}\left(\frac{\mathrm{A}}{\mathrm{2}}\right){cos}\left(\frac{\mathrm{B}}{\mathrm{2}}\right){cos}\left(\frac{\mathrm{C}}{\mathrm{2}}\right)\right)=? \\ $$$$\:\:\:\:\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:…