Question Number 135525 by mnjuly1970 last updated on 13/Mar/21
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\:{nice}\:…………….\:{calculus}… \\ $$$$\:\:\:\:\:{evaluation}\:{of}\:::\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} {xe}^{β{x}} \sqrt{\mathrm{1}β{e}^{β{x}} }\:{dx} \\ $$$$\:\:\:\:{solution}::\: \\ $$$$\:\:\:\:\mathrm{1}β{e}^{β{x}} ={t}\:\:\Rightarrow\:\left\{_{\:{x}=β{ln}\left(\mathrm{1}β{t}\right)} ^{\:{e}^{β{x}} {dx}={dt}} \right. \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}=β\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\mathrm{1}β{t}\right).{t}^{\frac{\mathrm{1}}{\mathrm{2}}} {dt} \\ $$$$\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{t}^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} }{{n}}{dt}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\:…. \\ $$$$\:\:\:\:\:\:\:\:\therefore\:\:\boldsymbol{\phi}=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\frac{\mathrm{3}}{\mathrm{2}}\right)}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}β\frac{\mathrm{1}}{{n}+\frac{\mathrm{3}}{\mathrm{2}}}\:…. \\ $$$$\:\:\:\:\:\:=\frac{\mathrm{2}}{\mathrm{3}}\left\{\gammaβ\gamma+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}}β\frac{\mathrm{1}}{{n}+\frac{\mathrm{3}}{\mathrm{2}}}\right)\right\}\:…. \\ $$$$\:\:\:\:\:{we}\:{know}\:{that}\::\: \\ $$$$\:\:\:\:\:\:\:\psi\left({s}+\mathrm{1}\right)\::=\:β\gamma+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}}β\frac{\mathrm{1}}{{n}+{s}}\right)\:….. \\ $$$$\:\:\:\:\:\:\:\:\:\therefore\:\boldsymbol{\phi}=\frac{\mathrm{2}}{\mathrm{3}}\left(\gamma+\psi\left(\frac{\mathrm{5}}{\mathrm{2}}\right)\right)\:…. \\ $$$$\:\:\:\:\:\:\:\:\:{on}\:{the}\:{oyher}\:{hand}\:{we}\:{have}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\psi\left({s}+\mathrm{1}\right)=\frac{\mathrm{1}}{{s}}+\psi\left({s}\right)\:…… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=β\gammaβ\mathrm{2}{ln}\left(\mathrm{2}\right)\:…… \\ $$$$\:\:\:\:\:\:\:\:\:\psi\left(\frac{\mathrm{5}}{\mathrm{2}}\right)=\frac{\mathrm{2}}{\mathrm{3}}+\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\mathrm{2}}{\mathrm{3}}+\left(\mathrm{2}+\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{2}+\left(β\gammaβ\mathrm{2}{ln}\left(\mathrm{2}\right)\right)\:…. \\ $$$$\:\:\:\:\:\:\:\:\therefore\:\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\mathrm{8}}{\mathrm{3}}β\gammaβ{ln}\left(\mathrm{4}\right)\:…. \\ $$$$\:\:\:\:\:\:\:\:\:::::::\:\:\:\:\:\boldsymbol{\phi}\:=\frac{\mathrm{2}}{\mathrm{3}}\left(\gamma+\frac{\mathrm{8}}{\mathrm{3}}β\gammaβ{ln}\left(\mathrm{4}\right)\right)\:…. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{8}}{\mathrm{3}}β{ln}\left(\mathrm{4}\right)\right)=\frac{\mathrm{4}}{\mathrm{3}}\left(\frac{\mathrm{4}}{\mathrm{3}}β{ln}\left(\mathrm{2}\right)\right)\:…. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……\:\boldsymbol{\phi}=\:\frac{\mathrm{4}}{\mathrm{3}}\left(\frac{\mathrm{4}}{\mathrm{3}}β{ln}\left(\mathrm{2}\right)\right)……\checkmark\checkmark\:…. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{prepare}\:{by}\:{mr}\:{rizzy}β{aka}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{solution}\:{with}\:{detais}\::{m}.{n}.{july}.\mathrm{1970} \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\: \\ $$$$ \\ $$
Commented by Dwaipayan Shikari last updated on 13/Mar/21
$$\delta\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}β{t}\right)^{{a}β\mathrm{1}} {dt}\Rightarrow\frac{\partial\delta\left({a}\right)}{\partial{a}}=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{1}β{t}\right)^{{a}β\mathrm{1}} {log}\left(\mathrm{1}β{t}\right){dt} \\ $$$$\frac{\partial\delta\left({a}\right)}{\partial{a}}=\frac{\partial}{\partial{a}}\left(\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left({a}\right)}{\Gamma\left({a}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\right)=\frac{\sqrt{\pi}}{\mathrm{2}}\left(\frac{\Gamma\left({a}+\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma'\left({a}\right)β\Gamma'\left({a}+\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left({a}\right)}{\Gamma^{\mathrm{2}} \left({a}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\right) \\ $$$${a}=\mathrm{1} \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{2}}\left(\frac{β\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}}\gammaβ\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}}\left(β\gamma+\frac{\mathrm{8}}{\mathrm{3}}β\mathrm{2}{log}\left(\mathrm{2}\right)\right)}{\frac{\mathrm{9}\pi}{\mathrm{16}}}\right) \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\left({log}\left(\mathrm{4}\right)β\frac{\mathrm{8}}{\mathrm{3}}\right) \\ $$$$β\delta'\left(\mathrm{1}\right)=\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{8}}{\mathrm{3}}β{log}\left(\mathrm{4}\right)\right) \\ $$
Commented by mnjuly1970 last updated on 13/Mar/21
$${thanking}\:{mr}\:{payan}… \\ $$