How to calculate this n dimensional integral?

The integral such as enter image description here

It’s easy to evaluate the first few items

Integrate[x1 (1/(-1+n))^n (1/v1)^(n/(-1+n)) x1^((2-n)/(-1+n)), {x1, 0, v1}] Integrate[x1 Integrate[(1/(-1+n))^n (1/v1)^(n/(-1+n)) x1^((2-n)/(-1+n)) x2^((2-n)/(-1+n)), {x2, 0, x1}], {x1, 0, v1}] Integrate[x1 Integrate[Integrate[(1/(-1+n))^n (1/v1)^(n/(-1+n)) x1^((2-n)/(-1+n)) x2^((2-n)/(-1+n)) x3^((2-n)/(-1+n)), {x3, 0, x1}], {x2, 0, x1}], {x1, 0, v1}] 

but how do I calculate this integral where the number of integrals is aribtrary?

finite dimensional modules are highest weight modules

Let $ \mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $ \mathfrak{g}$ has a highest weight vector.

My feeling is, since $ e_i$ ‘s are rising operators it will kill a non-zero vector and this will give us a highest weight vector and may be we need to use Lie’s theorem.

But I am unable to connect these things to get a perfect answer. If some one can tell me clearly what is happening here, that would help me a lot. Thank you.