Dear Stackexchange community,
I am studying the following integral as a function of parameter $a$: $$I(a)=\int_1^\infty e^{-\frac{1+x}{2a}}\frac{1}{a\sqrt{x}{(1-erf(\frac{1}{\sqrt{2a}})})^2}(erf(\frac{x}{\sqrt{2}})-erf(\frac{1}{\sqrt{2}}))dx$$ where $erf$ is the error function $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$.
I am trying to prove that $I(a)$ is decreasing in $a$. All sorts of numerical analysis confirms that this is so. Also, the nature of the problem where this comes from suggests that it should be the case (it is a density of a truncated distribution multiplied by another density).
The function under the integral looks as follows:
I wonder what techniques exist to prove such a fact. Taking a derivative doesn't seem to help: it is a messy integral of a function that is sometimes positive and sometimes negative.
Thank you for any advice.