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x-x-dx-




Question Number 107433 by Adilali last updated on 10/Aug/20
∫x^x dx=?
$$\int{x}^{{x}} {dx}=? \\ $$
Answered by Dwaipayan Shikari last updated on 10/Aug/20
∫e^(xlogx) dx  ∫e^t (1+logx).(1/((1+logx)))dx                                   xlogx=t  ,  1+logx=(dt/dx)                                                                                            logxe^(logx) =t , logx=W(t)  ∫(e^t /(1+logx))dt                             ∫(e^t /(1+W(t)))dt.....
$$\int{e}^{{xlogx}} {dx} \\ $$$$\int{e}^{{t}} \left(\mathrm{1}+{logx}\right).\frac{\mathrm{1}}{\left(\mathrm{1}+{logx}\right)}{dx}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{xlogx}={t}\:\:,\:\:\mathrm{1}+{logx}=\frac{{dt}}{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{logxe}^{{logx}} ={t}\:,\:{logx}={W}\left({t}\right) \\ $$$$\int\frac{{e}^{{t}} }{\mathrm{1}+{logx}}{dt}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\int\frac{{e}^{{t}} }{\mathrm{1}+{W}\left({t}\right)}{dt}….. \\ $$

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