Question Number 188380 by Shlock last updated on 28/Feb/23
Answered by SEKRET last updated on 28/Feb/23
![f(t)=β«_0 ^( π) ((ln(1+tβsin(a)βcos(x)))/(cos(x)))dx f β² (t)= β«_0 ^( π) ((sin(a)βcos(x))/(cos(x)β(1+tβsin(a)βcos(x))))dx f β² (t)= sin(a)ββ«_0 ^( π) (1/(1+tβsin(a)βcos(x)))dx sin(a) β[((2arctg(((tg((x/2))β(tβsin(a)β1))/( (β(1βt^2 sin^2 (a) ))))))/( (β(1βt^2 βsin^2 (a))) ))]_0 ^( π) =sin(a)β((π/( (β(1βt^2 βsin^2 a))))) f(t)= πβsin(a)ββ« (1/( (β(1βt^2 sin^2 a))))dt f(t)=πβsin(a)β((arcsin(tβsin(a)))/(sin(a))) t=1 I = πβa β«_0 ^( π) ((ln(1+sin(a)βcos(x)))/(cos(x)))dx= πβa ABDULAZIZ ABDUVALIYEV](https://www.tinkutara.com/question/Q188388.png)
$$\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\centerdot\left(\mathrm{1}+\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{f}}\:'\:\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\:\centerdot\left[\frac{\mathrm{2}\boldsymbol{\mathrm{arctg}}\left(\frac{\boldsymbol{\mathrm{tg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)\centerdot\left(\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)β\mathrm{1}\right)}{\:\sqrt{\mathrm{1}β\boldsymbol{\mathrm{t}}^{\mathrm{2}} \boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{a}}\right)\:}}\right)}{\:\sqrt{\mathrm{1}β\boldsymbol{\mathrm{t}}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{a}}\right)}\:}\right]_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \\ $$$$\:\:=\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\left(\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{1}β\boldsymbol{\mathrm{t}}^{\mathrm{2}} \centerdot\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{a}}}}\right) \\ $$$$\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\int\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}β\boldsymbol{\mathrm{t}}^{\mathrm{2}} \boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{a}}}}\boldsymbol{\mathrm{dt}} \\ $$$$\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{t}}\right)=\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\frac{\boldsymbol{\mathrm{arcsin}}\left(\boldsymbol{\mathrm{t}}\centerdot\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)} \\ $$$$\:\:\boldsymbol{\mathrm{t}}=\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{I}}\:=\:\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{a}} \\ $$$$\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \:\frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}+\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{a}}\right)\centerdot\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)}{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}\boldsymbol{\mathrm{dx}}=\:\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{a}} \\ $$$$\:\boldsymbol{{ABDULAZIZ}}\:\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$$$ \\ $$