`For example:`

Suppose we have an impulsive force $f(t)$ lasting from $t=t\_0$ until $t=t\_1$ which is applied to a mass $m$. Then by Newtons Second law we have $$\int\_{t=t\_0}^{t=t\_1}f(t)\,\mathrm{d}t=\int\_\color{red}{t=t\_0}^\color{red}{t=t\_1}m\color{blue}{\frac{\mathrm{d}v}{\mathrm{d}t}}\mathrm{d}t=\int\_\color{red}{v=v\_0}^\color{red}{v=v\_1}m\,\mathrm{d}v=m(v\_1-v\_0)\tag{1}$$

What I can't understand is; What substitution was made to allow the limits marked $\color{red}{\mathrm{red}}$ to change from $t$ to $v$. 

I thought it might be due to $$\color{blue}{\frac{\mathrm{d}v}{\mathrm{d}t}}=\frac{\mathrm{d}v}{\mathrm{d}x}\cdot \underbrace{\frac{\mathrm{d}x}{\mathrm{d}t}}\_{\Large{\color{#180}{=v}}}=v\frac{\mathrm{d}v}{\mathrm{d}x}$$ by the chain rule.

But it is something much simpler than this, and I believe I am over-thinking it too much. Could someone please tell me what substitution was made to change the limits marked $\color{red}{\mathrm{red}}$ in equation $(1)$?

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Edit:
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Comments below seem to indicate that one can simply change the limits of integration to make the integral dimensionally correct. But I consider this to be a less rigorous approach, and I was taught that integral limits **must** be changed via a substitution. So I still need to know what substitution was made?

Thanks again.