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I am trying to solve some exercises from Introduction to Differential Topology by Brocker and Janich. In particular, the exercise 8.13.2 requires me to show that for any vector field $X$ on an arbitrary manifold $M$ there exists a positive smooth function $f:M\to \mathbb{R}_+$ such that $fX$ is globally integrable.

My idea was to pick an exhaustion of the manifold by compact sets and to try to construct $f$ as a sum of appropriately weighted bump functions. However, I am not sure this leads anywhere or at least I am not able to set up the construction correctly.

I have seen some related questions here, but none of them answers this exact question (in particular, $X$ can vanish and is not complete to begin with). Any hint will be appreciated.

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    By "globally integrable" do you mean that all orbits are global? – 2017-02-27
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    By globally integrable (aka complete) vector field I mean that its flow exists for all times. To put it differently the domain of definition of the flow is equal to $\mathbb{R}\times M$. – 2017-02-27
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    Then you only need to make sure that $f$ approaches zero sufficiently fast near the boundary. – 2017-02-27
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    Ok, then my original idea might still be applicable. I will try to think about it some more. Thank you for the hint. – 2017-02-28

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