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Q-210956-im-read-leithold-book-again-in-this-book-1-define-ln-x-1-x-dx-x-x-gt-0-2-define-ln-e-1-1-e-dx-x-3-define-exp-x-y-ln-y-x-d-ln-u-du-1-u-d-ln-u-dx

Question Number 210967 by mahdipoor last updated on 24/Aug/24 $${Q}.\mathrm{210956} \\ $$$${im}\:{read}\:{leithold}\:{book}\:{again}\:,\:{in}\:{this}\:{book}\:: \\ $$$$\left.\mathrm{1}\right\}{define}\::\:{ln}\left({x}\right)=\int_{\mathrm{1}} ^{\:{x}} {dx}/{x}\:\:\:\:\:\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right\}{define}\::\:{ln}\left({e}\right)=\mathrm{1}=\int_{\mathrm{1}} ^{\:{e}} {dx}/{x} \\ $$$$\left.\mathrm{3}\right\}{define}\::\:{exp}\left({x}\right)={y}\:\Leftrightarrow\:{ln}\left({y}\right)={x} \\ $$$$\frac{{d}\left({ln}\left({u}\right)\right)}{{du}}=\frac{\mathrm{1}}{{u}}\:\Rightarrow\:\frac{{d}\left({ln}\left({u}\right)\right)}{{dx}}=\frac{{du}/{dx}}{{u}}\:\Rightarrow \\…

show-that-0-x-e-xt-e-t-2-dt-e-x-2-4-x-0-e-t-2-4-

Question Number 210839 by klipto last updated on 19/Aug/24 $$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\int_{\mathrm{0}} ^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{xt}}} \boldsymbol{\mathrm{e}}^{−\mathrm{t}^{\mathrm{2}} } \boldsymbol{\mathrm{dt}}=\boldsymbol{\mathrm{e}}^{\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{4}}} \underset{\mathrm{0}} {\int}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{e}}^{−\frac{\boldsymbol{\mathrm{t}}^{\mathrm{2}} }{\mathrm{4}}} \\ $$$$ \\ $$ Commented…