For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$.
However, in the context of the Atiyah-Singer Index Theorem one considers the fundamental class of the tangent bundle, $[TX]$ and evaluates a certain compactly support cohomology class on it to obtain the index. The class $[TX]$ does not seem to be well-defined in the sense described above.
Here is one reference from Michelsohn and Lawson's Spin Geometry, p.254:
Recall that for any manifold $X$, the tangent bundle $TX$ is canonically an almost complex manifold since $T(TX) = \pi^*TX \oplus \pi^*TX \cong \pi^*TX \otimes \mathbb{C}$. This gives $TXT$ a canonical orientation as a manifold. A positively oriented basis of $T(TX)$ is of the form $(e_1, Je_1,e_2,Je_2,\dots, e_n,Je_n)$ where $e_1, \dots, e_n$ is a basis of $\pi^*TX$ and $J$ carries the "horizontal" to the "vertical" factor. With this orientation we can evaluate any element $u \in H^{2n}_{cpt}(TX)$ on the fundamental class $[TX]$ of the manifold. The result is denoted by $u[TX]$.
So it certainly meant to be understood in the usual way. Hence, my question is: What is meant by $[TX]$?
My guess is that it actually it supposed to mean that we interpret cohomology in terms of differential form and then integrate over $TX$, but the way it is stated above does not seem to make any sense.
I'm aware of the question Fundamental class of cotangent bundle, but it didn't contain references and consequently didn't receive an answer.