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Question Number 76963 by peter frank last updated on 01/Jan/20 $$\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\left(\frac{\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}\right) \\ $$ Answered by MJS last updated on 02/Jan/20 $$\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\frac{\mathrm{cos}\:\theta\:+\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta\:−\mathrm{sin}\:\theta}\:{d}\theta= \\ $$$$=\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\frac{\mathrm{cos}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{sin}\:\mathrm{2}\theta}\:{d}\theta= \\ $$$$\:\:\:\:\:\mathrm{by}\:\mathrm{parts}…
Question Number 76965 by peter frank last updated on 02/Jan/20 $${Evaluate} \\ $$$${I}_{{ab}} =\int\mathrm{sin}\:{ax}\mathrm{cos}\:{bxdx} \\ $$$${if}\:{a}\neq{b}\:{and}\:{use}\:{it}\:{to} \\ $$$$\int_{\mathrm{0}} ^{{n}} \mathrm{sin}\:\mathrm{3}{x}\mathrm{cos}\:\mathrm{2}{xdx}=\frac{\mathrm{3}−\sqrt{\mathrm{3}}}{\mathrm{5}} \\ $$ Answered by mr…
Question Number 11399 by tawa last updated on 23/Mar/17 $$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{and}\:\mathrm{last}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}\:\mathrm{is}\:\mathrm{51}.\:\mathrm{And}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{progression}\:\mathrm{is}\:\mathrm{255}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{A}.\mathrm{P}. \\ $$ Answered by ajfour last updated on 23/Mar/17 $$\mathrm{last}\:\mathrm{term}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{51}+\mathrm{9d}\right) \\ $$$$\mathrm{where}\:\mathrm{d}\:\mathrm{is}\:\mathrm{whatever}\:\mathrm{common} \\…
Question Number 76929 by peter frank last updated on 01/Jan/20 $$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}\left(\mathrm{sin}\left({a}+{x}\right)\right)\:}} \\ $$ Answered by MJS last updated on 01/Jan/20 $$\int\frac{{dx}}{\:\sqrt{\mathrm{sin}^{\mathrm{3}} \:{x}\:\mathrm{sin}\:\left({a}+{x}\right)}}= \\ $$$$=\int\frac{{dx}}{\:\sqrt{\mathrm{sin}^{\mathrm{3}}…
Question Number 76872 by john santu last updated on 31/Dec/19 $${the}\:{product}\:{of}\:\mathrm{3}\:{integer}\:{x},{y},{z}\:{is}\:\mathrm{192} \\ $$$$.\:{z}\:=\:\mathrm{4}\:{and}\:{t}\:{is}\:{equal}\:{to}\:{average}? \\ $$$${of}\:{x}\:{and}\:{y}\:.\:{what}\:{is}\:{the}\:{minimum}\: \\ $$$${posible}\:{value}\:{of}\:{t}? \\ $$ Commented by mr W last updated…
Question Number 76830 by peter frank last updated on 30/Dec/19 Commented by mathmax by abdo last updated on 05/Jan/20 $${let}\:{remember}\:{that}\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right)\:\left({result}\:{proved}\right) \\ $$$${iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\:=−{iln}\left(\frac{{a}+{ib}}{{a}−{ib}}\right)\:=−{iln}\left(\frac{\mathrm{1}+{i}\frac{{b}}{{a}}}{\mathrm{1}−{i}\frac{{b}}{{a}}}\right)\:=−{i}\left(\mathrm{2}{i}\right){arctan}\left(\frac{{b}}{{a}}\right) \\ $$$$=\mathrm{2}\:{arctan}\left(\frac{{b}}{{a}}\right)\:\Rightarrow{tan}\left({iln}\left(\frac{{a}−{ib}}{{a}+{ib}}\right)\right)={tan}\left(\mathrm{2}{arctan}\left(\frac{{b}}{{a}}\right)\right) \\…
Question Number 76718 by peter frank last updated on 29/Dec/19 $${For}\:{n}\:\in\:{N}\:{prove}\:{by}\:{mathematical} \\ $$$${induction}\:{that} \\ $$$$\mathrm{cos}\:\alpha+\mathrm{cos}\:\left(\alpha+\beta\right)+\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]+…\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]= \\ $$$$\frac{\mathrm{cos}\:\left[\alpha+\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)\beta\right]\mathrm{sin}\:\frac{{n}\beta}{\mathrm{2}}}{\mathrm{sin}\:\frac{{n}}{\mathrm{2}}} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 76716 by peter frank last updated on 29/Dec/19 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$ Commented by abdomathmax last updated on…
Question Number 76717 by peter frank last updated on 29/Dec/19 $${A}\:{triangle}\:{is}\:{formed}\:{by} \\ $$$${the}\:{three}\:{straight}\:{line} \\ $$$${y}={m}_{\mathrm{1}} {x}+\frac{{a}}{{m}_{\mathrm{1}} } \\ $$$${y}={m}_{\mathrm{2}} {x}+\frac{{a}}{{m}_{\mathrm{2}} } \\ $$$${y}={m}_{\mathrm{3}} {x}+\frac{{a}}{{m}_{\mathrm{3}} }…