The dual of a space of compact operators

Let $X$ and $Y$ be Banach spaces. We provide the representation of the dual space of compact operators $K(X,Y)$ as a subspace of bounded linear operators $\mathcal{L}(X,Y)$. The main results are: (1) If $Y$ is separable, then the dual forms of $K(X,Y)$ can be represented by the integral operator and the elements of $C[0,1]$. (2) If $X^{\ast \ast }$ has the weak Radon-Nikodym property, then the dual forms of $K(X,Y)$ can be represented by the trace of some tensor products.