Question Number 172653 by Mikenice last updated on 29/Jun/22
Answered by MJS_new last updated on 30/Jun/22
![∫(((−1)^(1/x) )/x)dx= [t=((πi)/x) → dx=((ix^2 )/π)dt] =−∫(e^t /t)dt=−Ei (t) = =−Ei (((iπ)/x)) +C](Q172676.png)
$$\int\frac{\left(−\mathrm{1}\right)^{\mathrm{1}/{x}} }{{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\pi\mathrm{i}}{{x}}\:\rightarrow\:{dx}=\frac{\mathrm{i}{x}^{\mathrm{2}} }{\pi}{dt}\right] \\ $$$$=−\int\frac{\mathrm{e}^{{t}} }{{t}}{dt}=−\mathrm{Ei}\:\left({t}\right)\:= \\ $$$$=−\mathrm{Ei}\:\left(\frac{\mathrm{i}\pi}{{x}}\right)\:+{C} \\ $$