Background
I'm reading Hornung (1997)'s Homogenization and porous media, pg 3:
We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, \begin{align}\frac{d}{dx}\left( a^\epsilon(x)\frac{du^\epsilon}{dx}\right)=0 \text{ in (0,1)}, \\ u^\epsilon(x)=0 \text{ at } x=0 \\ u^\epsilon(x)=1 \text{ at } x=1 \end{align}
where we make the assumption that the functions $a^\epsilon(.)$ are given in the form $a^\epsilon(x)=a(\frac{x}{\epsilon}$ with a fixed periodic function $a(.)$ having periodicity of length $1$. The meaning of this is that the functions $a^\epsilon(.)$ are oscillatory with increasing frequencies for $\epsilon\rightarrow 0$. [In this case], we get
\begin{align} u^\epsilon(x)=\frac{\int_0^x \frac{dt}{a^\epsilon(t)} }{\int_0^1 \frac{dt}{a^\epsilon(t)}} = \frac{\int_0^{nx}\frac{dy}{a(y)}}{\int_0^n \frac{dy}{a(y)} } \end{align}
I thoroughly understand how the author obtained this result. However...
My Question
The author claims claims that
\begin{align} \frac{\int_0^{nx}\frac{dy}{a(y)}}{\int_0^n \frac{dy}{a(y)} } = x + \frac{ \int_0^{nx} \left(\frac{1}{a(y)} -1 \right) dy }{\int_0^n \frac{dy}{a(y)}} \end{align}
However, I cannot seem to verify this identity. I've tried simplifying the RHS, in which case I obtain
\begin{eqnarray} x + \frac{ \int_0^{nx} \left(\frac{1}{a(y)} -1 \right) dy }{\int_0^n \frac{dy}{a(y)}} & = & x + \frac{ \int_0^{nx} \frac{1}{a(y)} dy }{\int_0^n \frac{dy}{a(y)}} - \frac{ \int_0^{nx} 1 dy }{\int_0^n \frac{dy}{a(y)}} \\ & = & x + \frac{ \int_0^{nx} \frac{1}{a(y)} dy }{\int_0^n \frac{dy}{a(y)}} - \frac{ nx}{\int_0^n \frac{dy}{a(y)}} \end{eqnarray}
I notice that the middle term in this last expression
$$\frac{ \int_0^{nx} \frac{1}{a(y)} dy }{\int_0^n \frac{dy}{a(y)}}$$
is exactly equal to what I wanted to verify. However, the two other terms $x$ and $-\frac{ nx}{\int_0^n \frac{dy}{a(y)}}$ do not necessarily cancel.
I'm getting the feeling that either I'm missing something totally obvious or that this so-called 'identity' is a typo (I doubt the latter).
Any help verifying this identity would be greatly appreciated.
Thanks.