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Question-188380




Question Number 188380 by Shlock last updated on 28/Feb/23
Answered by SEKRET last updated on 28/Feb/23
  f(t)=∫_0 ^( 𝛑) ((ln(1+tβˆ™sin(a)βˆ™cos(x)))/(cos(x)))dx   f β€² (t)= ∫_0 ^( 𝛑)  ((sin(a)βˆ™cos(x))/(cos(x)βˆ™(1+tβˆ™sin(a)βˆ™cos(x))))dx   f β€² (t)= sin(a)βˆ™βˆ«_0 ^( 𝛑) (1/(1+tβˆ™sin(a)βˆ™cos(x)))dx   sin(a) βˆ™[((2arctg(((tg((x/2))βˆ™(tβˆ™sin(a)βˆ’1))/( (√(1βˆ’t^2 sin^2 (a) ))))))/( (√(1βˆ’t^2 βˆ™sin^2 (a))) ))]_0 ^( 𝛑)     =sin(a)βˆ™((𝛑/( (√(1βˆ’t^2 βˆ™sin^2 a)))))   f(t)= π›‘βˆ™sin(a)βˆ™βˆ« (1/( (√(1βˆ’t^2 sin^2 a))))dt    f(t)=π›‘βˆ™sin(a)βˆ™((arcsin(tβˆ™sin(a)))/(sin(a)))    t=1            I =  π›‘βˆ™a   ∫_0 ^( 𝛑)  ((ln(1+sin(a)βˆ™cos(x)))/(cos(x)))dx= π›‘βˆ™a   ABDULAZIZ   ABDUVALIYEV
$$\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\centerdot\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\:\centerdot\left[\frac{\mathrm{2}\boldsymbol{\mathrm{arctg}}\left(\frac{\boldsymbol{\mathrm{tg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)\centerdot\left(\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)βˆ’\mathrm{1}\right)}{\:\sqrt{\mathrm{1}βˆ’\boldsymbol{\mathrm{t}}^{\mathrm{2}} \boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{a}}\right)\:}}\right)}{\:\sqrt{\mathrm{1}βˆ’\boldsymbol{\mathrm{t}}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{a}}\right)}\:}\right]_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \\ $$$$\:\:=\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\left(\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{1}βˆ’\boldsymbol{\mathrm{t}}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{a}}}}\right) \\ $$$$\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}βˆ’\boldsymbol{\mathrm{t}}^{\mathrm{2}} \boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{a}}}}\boldsymbol{\mathrm{dt}} \\ $$$$\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\frac{\boldsymbol{\mathrm{arcsin}}\left(\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)} \\ $$$$\:\:\boldsymbol{\mathrm{t}}=\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{I}}\:=\:\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{a}} \\ $$$$\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}}=\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{a}} \\ $$$$\:\boldsymbol{{ABDULAZIZ}}\:\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$$$ \\ $$

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