Let $(X_1,X_2)$ be a random vector with $X_1 \sim Exp(1)$ and $X_2 \sim N(0,1)$. The dependence structure is given by the copula $$C(u_1,u_2)=\frac{1}{3}W(u_1,u_2)+\frac{2}{3}\Pi(u_1,u_2), \ \ \ u \in [0,1]^2$$ where
$W(u_1,u_2):=(u_1+u_2-1)_+$ is the contramonotone copula and
$\Pi(u_1,u_2):=u_1u_2$ is the independence copula.
I want to evaluate Kendall's Tau $p_\tau(X_1,X_2)$. We got the following theorem:
$$p_\tau(X_1,X_2)=4\int_0^1\int_0^1C(u_1,u_2)dC(u_1,u_2)-1$$
We have $C(u_1,u_2)=\frac{1}{3}(u_1+u_2-1)_+\frac{2}{3}u_1u_2$.
What is $dC(u_1,u_2)?$ And what are our limits of integration?
Maybe it works like here: Clayton copula and Kendall's tau but I am not sure