Let $X_t=\int_0^t \left(\frac{1-t}{1-u}\right)^k dW_u$. Assume $0\lt s \lt t\lt T$. Is the following the right way to compute the covariation of $X_s$ and $X_t$?
$$ \begin{align} \text{Cov}(X_t, X_s) &= \mathbb{E}\left[\left( X_t - \mathbb{E}[X_t] \right)\left( X_s - \mathbb{E}[X_s] \right)\right]\\ &=\mathbb{E}[X_t X_s]\\ &=\mathbb{E}\left[ \int_0^t \left(\frac{1-t}{1-u}\right)^k dW_u\int_0^s \left(\frac{1-s}{1-u}\right)^k dW_u\right]\\ &=\mathbb{E}\left[ \left(\int_s^t \left(\frac{1-t}{1-u}\right)^k dW_u+\int_0^s \left(\frac{1-t}{1-u}\right)^k dW_u\right)\int_0^s \left(\frac{1-s}{1-u}\right)^k dW_u\right]\\ &=\mathbb{E}\left[ \int_0^s \left(\frac{1-t}{1-u}\right)^k dW_u\int_0^s \left(\frac{1-s}{1-u}\right)^k dW_u\right]\\ &=\mathbb{E}\left[ \int_0^s \left(\frac{1-t}{1-u}\right)^k\left( \frac{1-s}{1-u}\right)^k du\right]\\ \end{align} $$