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A function $f:{\Omega}{\rightarrow}X$ is called intrinsically-separable valued if there exists $E{\in}{\Sigma}$ with ${\mu}(E)=0$ such that $f({\Omega}-E)$ is a separable in X. For a given Dunford integrable function $f:{\Omega}{\rightarrow}X$ and a weakly compact operator T, we show that if f is intrinsically-separable valued, then f is Pettis integrable, and if there exists a sequence ($f_n$) of Dunford integrable and intrinsically-separable valued functions from ${\Omega}$ into X such that for each $x^*{\in}X^*$, $x^*f_n{\rightarrow}x^*f$ a.e., then f is Pettis integrable. We show that a function f is Pettis integrable if and only if for each $E{\in}{\Sigma}$, F(E) is $weak^*$-continuous on $B_{X*}$ if and only if for each $E{\in}{\Sigma}$, $M=\{x^*{\in}X^*:F(E)(x^*)=O\}$ is $weak^*$-closed.
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