Question Number 140399 by mnjuly1970 last updated on 07/May/21 | ||
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\xi}\::=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{β{x}^{\mathrm{2}} } β{e}^{β{x}} }{{x}}\:{dx}\:=\:{k}.\gamma\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{find}\:\:''\:{k}\:\:''\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\gamma\::=\:\mathscr{E}{uler}\:{constant}.... \\ $$ | ||
Answered by qaz last updated on 07/May/21 | ||
$$β\gamma=\int_{\mathrm{0}} ^{\infty} \left({e}^{β{x}} β\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)\frac{{dx}}{{x}} \\ $$$$β\gamma=\int_{\mathrm{0}} ^{\infty} \left({e}^{β{x}^{\mathrm{2}} } β\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)\frac{{d}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \left({e}^{β{x}^{\mathrm{2}} } β\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)\frac{{dx}}{{x}} \\ $$$$\Rightarrowβ\frac{\gamma}{\mathrm{2}}=\int_{\mathrm{0}} ^{\infty} \left({e}^{β{x}^{\mathrm{2}} } β\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)\frac{{dx}}{{x}} \\ $$$$\xi=\int_{\mathrm{0}} ^{\infty} \left({e}^{β{x}^{\mathrm{2}} } β{e}^{β{x}} \right)\frac{{dx}}{{x}} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \left\{\left({e}^{β{x}^{\mathrm{2}} } β\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)β\left({e}^{β{x}} β\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)+\left(\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }β\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)\right\}\frac{{dx}}{{x}} \\ $$$$=β\frac{\gamma}{\mathrm{2}}+\gamma+\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }β\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)\frac{{dx}}{{x}} \\ $$$$=\frac{\gamma}{\mathrm{2}}+\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\mathrm{1}+{x}}β\frac{{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx} \\ $$$$=\frac{\gamma}{\mathrm{2}}+{ln}\frac{\mathrm{1}+{x}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\mid_{\mathrm{0}} ^{\infty} \\ $$$$=\frac{\gamma}{\mathrm{2}} \\ $$$$\Rightarrow{k}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$ | ||
Commented by mnjuly1970 last updated on 07/May/21 | ||
$$\:\:\:{bravo}\:\:...{mr}\:{payan}... \\ $$ | ||