I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be a geometric Brownian motion. Let $p$ be a positive constant. Compute $\mathrm{d}(\{S(t)\}^p)$, the differential of $S(t)$ raised to the power of $p$.
The details of how I solve this problem aren't too relevant to the question, so a reader can skip to The Question.
Solving the Problem
I can solve this problem with a direct application of the Itô-Doeblin formula (page 138). I reproduce that formula here for completeness.
Theorem 4.4.1 (Itô-Doeblin formula for Brownian motion). Let $f(t, x)$ be a function for which the partial derivatives $f_t(t, x)$, $f_x(t, x)$, and $f_{xx}(t, x)$ are defined and continuous, and let $W(t)$ be a Brownian motion. Then, for every $T\geq0$,
$$f(T, W(T)) = f(0,W(0)) + \int^{T}_0 f_t(t, W(t)) \mathrm{d}t + \int^{T}_0 f_x(t, W(t))\mathrm{d}W(t)+\frac{1}{2}\int^T_0f_{xx}(t, W(t))\mathrm{d}t\text{.}\tag{1}$$
We define
$$f(t, x) = \Big( S(0)e^{\sigma x + (\alpha-\frac{1}{2}\sigma^2)t}\Big)^p\text{.}\tag{2}$$
We then directly apply Theorem 4.4.1 to get \begin{align} \mathrm{d}f(t, W(t)) = \mathrm{d}(\{S(t)\}^p) &= pS(0)^p(\alpha+(\frac{p-1}{2})\sigma^2) e^{p(\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t)}\mathrm{d}t + pS(0)^p(\sigma)e^{p(\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t)}\mathrm{d}W(t) \\ &= pS(t)^p\Big[\sigma \mathrm{d}W(t) + (\alpha + \frac{p-1}{2})\mathrm{d}t \Big]\tag{3}\end{align}
or, to put it in integral form,
$$\int_0^T (S(t))^p \mathrm{d}t = \int_0^T pS(t)^p\sigma \mathrm{d}W(t) + \int_0^T pS(t)^p(\alpha + \frac{p-1}{2})\mathrm{d}t\text{.}\tag{4}$$
The Question
The math involved in the solution above makes sense, and I believe this solution matches the author's intent. However, I don't see why the above work has been productive. When I "compute" something, I imagine that you execute an algorithm to solve a problem (perhaps just applying the Itô-Doeblin formula was the "computation"), or, in most cases, put a problem in a "closed-form" where known algorithms can then solve it. Often you simplify a formula to remove complicating parts.
I don't see why I am done when I write line $(4)$. I am not sure how you would compute $(4)$ or ever use $(4)$. I am not sure why $(4)$ is a better form than just writing $\mathrm{d}\{S(t)\}^p)$. Assuming you had some interest in $\mathrm{d}\{S(t)\}^p)$, how would you ever use $(4)$? What would be the inputs? What would be the outputs? Would you use it with a computer, or just paper and pencil? Is there any use of this formula that isn't just getting a distribution of $\mathrm{d}\{S(t)\}^p)$ for time $t$ when you have all of the information up to $t-1$? Also, the book suggests integrals w.r.t. $\mathrm{d}t$ are "just Lebesgue" integrals, but when the integrand involves Brownian Motion, I don't see why that is helpful.
