Question: I have to solve this integral but I don't understand which substitution I have to use. I've tried with $\sqrt{v^2+w^2}=t$, $v^2+w^2=t$, adding and subtracting $1$ in the numerator, adding and subtracting $2w$ under square root, but no dice.

Answer: 
For convergence, we require $0 < |v| \leq 1$, and then without loss of generatlity we can take $v > 0$. The trick here is to let $w = v \sinh x$, so that
\begin{aligned}
v^2+w^2 &= v^2 (1 + \sinh^2 x) = v^2 \cosh^2 x, \\
dw &= v \cosh x \; dx, \\
x &= \sinh^{-1} \frac{w}{v}
\end{aligned}
Our integral then becomes
$$\int\_0^{\sinh^{-1} \sqrt{v^{-2}-1}} \frac{v \cosh x}{\sqrt{v^2 \cosh^2 x}} \; dx = \sinh^{-1} \sqrt{\frac{1}{v^2}-1}$$
since the integrand simplifies to become trivial.
Hint: recall that
$$ \int \frac{1}{\sqrt{1+w^2}} dw = \sinh^{-1}(w) +C $$
where $\sinh^{-1}$ indicates the inverse of the hyperbolic sine function. Consider the substitution $w=tv$ and proceed.