Let $1<p<\infty$ and $w(x):=x^{\lambda}$, where $\lambda>0$. One way to define the weighted Sobolev space $W^{1,p}((0,1),w)$ is the following (see [1]): we say that $u\in W^{1,p}((0,1),w)$ if $u$ has a weak (distributional) derivative $u^{\prime}$ and the norm $$ \big(\int_{0}^{1}\left|u(x)\right|^{p}w(x)\thinspace dx+\int_{0}^{1}\left|u^{\prime}(x)\right|^{p}w(x)\thinspace dx\big)^{1/p} $$ is finite. My question is that is $C^\infty [0,1]$ dense in this space?
Remark. In [1, Thm 7.4] it is shown that $C^\infty(\overline{\Omega})$ is dense in the space $W^{1,p}(\Omega,w)$, where $\Omega\subset\mathbb{\mathbb{R}}^{N}$ is a suitable domain and $w$ is given by $$w(x)=\left|x-x_{0}\right|^{\lambda}$$ for some fixed $\lambda\geq0$ and $x_{0}\in\partial\Omega$. This theorem would seemingly apply to my case, but the proof is complicated and I am unsure if it holds also when $\Omega$ is an interval.
[1] A. Kufner. Weighted Sobolev Spaces, Wiley, New York, 1985.