Does $\{\varphi(x)\in C_{c}^{\infty}(U)|\varphi(0)=0\}=\{x_{1}\psi_{1}+\cdots+x_{n}\psi_{n}|\psi_{i}\in C_{c}^{\infty}(U)\}?$

Question

This Exercise comes from https://math.mit.edu/~dyatlov/18.155/155-notes.pdf Please look at the picture, the hint about is strange for me? why should we show that $\mathcal{V}$ contains $C_{c}^{\infty}(U\setminus\{0\})$ by taking partition of unity? Why not directly to prove that $\varphi(x)=\varphi(0)+\sum_{j=1}^{n}\int_{0}^{1}\varphi_{x_{j}}(tx)dt\cdot x_{j}$. Does $\int_{0}^{1}\varphi_{x_{j}}(tx)dt\in C_{c}^{\infty}(U)$?

Any suggestions are wellcome!