I have a problem for to understand the following example of a distribution in the Lee's book of smooth manifolds:
If $V$ is a nowhere-vanishing smooth vector field on a manifold $M$, then $V$ spans a smooth rank-$1$ distribution $D$ on $M$. The image of any integral curve of $V$ is a integral manifold of $D$.
Well, as $V$ is non-vanishing, for each $p\in M$, then $D=span(V_p)$,i.e, is a set of straight lines passing in the origin is a rank-$1$ smooth distribution in $M$. I dont know to argue about the fact that the image of integral curves on $V$ are integral manifold.
The dimension of the integral manifold is $1$ since the rank of the distribution is $1$. An integral curve $c_t$ verifies ${d\over{dt}}c_t=X(c_t))$, so $X$ is tangent to the curve, and it is an integral manifold.