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Is it possible to have Mathematica evaluate the following integral?

$$I=\int_{0}^1 \dfrac{\sin\|a v\|}{\|a v\|} a \mathrm{d}a$$

where $v$ is a vector.

By hand: since $a\in[0,1]$ and with $\mu=\|a v\|=a\|v\|$:

$$I=\int_0^1 \dfrac{\sin(a\|v\|)}{\|v\|} \mathrm{d}a = \dfrac{1}{\|v\|^2}\int_0^{\|v\|} \sin(\mu))\mathrm{d}\mu = \dfrac{1-\cos(\|v\|)}{\|v\|^2}$$

The following does not evaluate:

Integrate[Sin[Norm[a*v]]*a/Norm[a*v], {a, 0, 1}]
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Slightly awkward, but certainly possible:

Assuming[v ∈ Vectors[n, Reals], 
         Integrate[TensorExpand[a Sin[Norm[a v]]/Norm[a v]], {a, 0, 1}]]
   (1 - Cos[Norm[v]])/Norm[v]^2
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