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Well, with questions like these, which are transformations, you often do something like this:
You know the t-distribution, I assume, so you isolate W for T (rearrange to get T). Since T is t-distro, you apply the t-distro pdf to this T, noting that T is a function of W. In this case, you want to find Pr(W<w) which is equal to G(w)=Pr(function of T). Now you simply differentiate G(w) to get the pdf of W, g(w).
Another method which may work for you is to isolate W for T, then plug this new T into the formula
g(w) = f(t) * abs(dt/dw), where f(t) is the t-distro pdf for the variable T=t.
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