Question Number 186531 by normans last updated on 05/Feb/23 $$ \\ $$$$\:\:\mathbb{Q}.\boldsymbol{{use}}\:\boldsymbol{{the}}\:\boldsymbol{{parseval}}\:\boldsymbol{{relation}}\:\boldsymbol{{of}}\:\boldsymbol{{hankel}}\:\boldsymbol{{transfrom}}\:\boldsymbol{{to}}\:\boldsymbol{{evaluate}}\:\boldsymbol{{the}}\:\boldsymbol{{Integral}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\infty} {\int}_{\mathrm{0}} \:\:\frac{\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{ar}}\right)\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{br}}\right)}{\boldsymbol{{r}}}\:,\:\:\boldsymbol{{for}}\:\boldsymbol{\gamma}>−\frac{\mathrm{1}}{\mathrm{2}}\:,\:\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}} \\ $$$$\:\:\:\boldsymbol{{where}}\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{x}}\right)\:\boldsymbol{{are}}\:\boldsymbol{{bessel}}\:\boldsymbol{{funtions}}. \\ $$$$ \\ $$…
Question Number 55457 by maxmathsup by imad last updated on 24/Feb/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx}\:. \\ $$ Commented by maxmathsup by imad last updated on…
Question Number 55454 by maxmathsup by imad last updated on 24/Feb/19 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+{a}}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intermsof}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{2}}{dx}…
Question Number 186527 by Mingma last updated on 05/Feb/23 Answered by ARUNG_Brandon_MBU last updated on 05/Feb/23 $${I}=\int\frac{\mathrm{cos}{x}}{\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{sin}{x}+\mathrm{1}}{dx}\:,\:{s}=\mathrm{sin}{x} \\ $$$$\:\:=\int\frac{{ds}}{{s}^{\mathrm{2}} +{s}+\mathrm{1}}=\int\frac{{ds}}{\left({s}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}} \\ $$$$\:\:=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{arctan}\left(\frac{\mathrm{2}{s}+\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{C} \\…
Question Number 120970 by Algoritm last updated on 04/Nov/20 Answered by mathmax by abdo last updated on 04/Nov/20 $$\mathrm{I}\:=\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{cosx}}{\mathrm{2}}\right)}{\mathrm{cosx}}\mathrm{dx}\:\:\:\mathrm{let}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{acosx}\right)}{\mathrm{cosx}}\mathrm{dx}\:\:\mathrm{with}\:\mid\mathrm{a}\mid<\mathrm{1} \\ $$$$\mathrm{I}\:=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\:\mathrm{we}\:\mathrm{hsve}\:\mathrm{f}^{'}…
Question Number 120964 by bramlexs22 last updated on 04/Nov/20 Commented by bramlexs22 last updated on 04/Nov/20 $$\mathrm{the}\:\mathrm{line}\:\mathrm{x}\:=\:\mathrm{3} \\ $$ Commented by liberty last updated on…
Question Number 186476 by Mingma last updated on 05/Feb/23 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 120898 by talminator2856791 last updated on 03/Nov/20 $$\: \\ $$$$\: \\ $$$$\: \\ $$$$\mathrm{evaluate}:\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{x}}{{x}+\mathrm{1}}\:−\:\underset{{k}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{k}}{{k}+\mathrm{1}}\right)^{\lfloor{k}\rfloor} \right)^{\lfloor{x}\rfloor} {dx} \\ $$$$\: \\ $$$$\:…
Question Number 55364 by peter frank last updated on 22/Feb/19 $$\mathrm{prove}\:\mathrm{that}\:\int_{−\infty\:} ^{\infty} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{1} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)}\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{n}\right)} \\ $$$$\mathrm{and}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\frac{\mathrm{n}}{\mathrm{2}}−\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\mathrm{dx} \\ $$ Terms…
Question Number 55360 by rahul 19 last updated on 22/Feb/19 $$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \sqrt{\mathrm{sin}\:\left(\mathrm{3}{x}−{x}^{\mathrm{2}} −\mathrm{2}\right)}{dx}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{3}} ^{\mathrm{1}} \sqrt{{sin}\left(\frac{\mathrm{4}{t}−{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{4}}\right)}{dt}\:\:=? \\ $$ Commented by rahul 19 last updated…