I am trying to understand the proof of Stokes' theorem found in Spivak's Calculus on Manifolds. However, I find the final lines are rather confusing:
$$ \begin{aligned} & \int_{I^k} \mathrm{d} (f dx^1 \wedge \ldots \wedge \widehat{dx^i} \wedge \ldots \wedge dx^k) \\ &= \int_{[0,1]^k} \frac{\partial f}{\partial x^i} dx^i \wedge dx^1 \wedge \ldots \wedge \widehat{dx^i} \wedge \ldots \wedge dx^k \\ &= (-1)^{i-1} \int_{I^k} \frac{\partial f}{\partial x^i} dx^1 \ldots dx^k \\ &= (-1)^{i-1} \int_0^1 \ldots \Big( \int_0^1 \frac{\partial f}{\partial x^i} (x^1, \ldots , x^k) dx^i \Big) dx^1 \ldots \widehat{dx^i} \ldots dx^k \\ &= (-1)^{i-1} \int_0^1 \ldots \int_0^1 (f(x^1, \ldots, 1, \ldots , x^k) - f(x^1, \ldots, 0, \ldots , x^k)) dx^1 \ldots \widehat{dx^i} \ldots dx^k \\ &= (-1)^{i-1} \int_{[0,1]^k} f(x^1, \ldots , 1, \ldots , x^k )dx^1 \ldots dx^k \\ &+ (-1)^i \int_{[0,1]^k} f(x^1, \ldots , 0, \ldots , x^k )dx^1 \ldots dx^k. \end{aligned} $$ I'm particularly confused about how he gets from $$ (-1)^{i-1} \int_0^1 \ldots \int_0^1 (f(x^1, \ldots, 1, \ldots , x^k) - f(x^1, \ldots, 0, \ldots , x^k)) dx^1 \ldots \widehat{dx^i} \ldots dx^k $$ to the last line where, unexpectedly, the $dx^i$ term reappears. I'm probably missing something very simple here but would appreciate any explanation for this.
Edit: My question actually boils down to why $$ \int_{[0,1]^{k-1}} f(x^1 , \ldots , \alpha, \ldots, x^k) dx^1 \wedge \ldots \wedge \widehat{dx^i} \wedge \ldots \wedge dx^k \\ = \int_{[0,1]^{k}} f(x^1 , \ldots , \alpha, \ldots, x^k) dx^1 \wedge \ldots \wedge dx^k ,$$ which I'm guessing has something to do with the fact that $$ \int_{0}^{1} dx^i = 1.$$