First things first, in the above graphs the one in blue is just a rotated version of ex from where the question did really form. I rotated it just in order to make comparisons easier.
If we did the integral of both the functions in [0,1] we see that it's value for the blue curve is finite(i.e. -0.632) but for the green one it is non-existent (of-course as 0 isn't in it's domain).
But again if we repeat the same process in (0,1] we find that although the value approaches the one we found before for the blue curve, it keeps on increasing (diverging) for the green one as closer and closer we get to 0.
I wish to know why even as both the curves are essentially of the same type in [0,1], one limits it's area to a finite value while the other keeps on diverging.
They are not of the same type: the blue curve is exponential decay. We have $$ \color{blue}{\int_{0}^{\infty} e^{-y}\,dy }= \left.- e^{-y}\right|_{0}^{\infty} = 1 - 0 =1 $$The green curve is a reciprocal function. We have $$ \color{green}{\int_{0}^{1} x^{-1}dx}= \left.\ln|x|\right|_{0}^{1} = 0 - (-\infty) =\infty $$