I have some trouble computing the integral in Exercise 4.3.1 in Bott and Tu; Differential Forms in Algebraic Topology, and I was wondering if someone could help me with that.
So we have the $n$-form $$\omega= \sum_{i=1}^{n}(-1)^{i-1}x_idx_1\ldots \widehat{dx_i}\ldots dx_{n+1}$$ (hat means omit) on the $n$ unit sphere $S^n$ and we have to integrate it over $S^n$. So, i did it over $S^1$ using the usual parametrization but higher dimensional analogues seem too complicated to compute. Is there an easier way? Thank you.
Note that $d\omega =(n+1)\ dx_1\cdots dx_{n+1}$ is a volume form on $\mathbb{R}^{n+1}$.
If $B$ is unit ball in $\mathbb{R}^{n+1}$, then $$ (n+1) {\rm vol}\ B = \int_B\ d\omega =\int_{S^n}\ \omega$$