Most books about manifolds say that "there is no way to integrate real-valued functions in a coordinate-independent way on a manifold". I've read the usual reasons but they don't seem to differ from the usual theory of integration in $\mathbb{R}^n$. I explain below what I mean.
Let $M$ be an orientable smooth $n$-manifold. Since it is orientable, we have a nonvanishing $n$-form $\omega$. This allows us to define the following linear functional: \begin{align*} \Lambda:C_c^{\infty}(M)&\to \mathbb{R}\\ f &\mapsto \int_M f\omega. \end{align*} By continuity, we can extend its domain to obtain a positive linear functional $C_c(M)\to \mathbb{R}$. Then, the Riesz-Markov-Kakutani representation theorem implies the existence of a regular Borel measure $\mu$ such that $$\Lambda(f)=\int_M f \:\mathrm{d}\mu,$$ for all $f\in C_c^\infty(M)$.
Why isn't this a good notion of integral of functions on a manifold? It doesn't seem to depend on a choice of chart and while it depends of a choice of $\omega$, I would argue that the same thing happens in $\mathbb{R}^n$ since the Lebesgue measure depends of a normalization factor.