I wish to make my integral symbols in pdfLaTeX documents appear the same way as in the following example:
The integral symbol comes from the Symbol font which is shipped with Microsoft products, and is made by combining the unicode symbols 2320, 2321, and 23AE. Is it possible to replicate this symbol in pdfLatex or a related typesetting engine? And if so, could anyone suggest potential ways that I might accomplish this?

unicode-mathwith XeTeX or LuaTeX. But don't hold your breath. Does the font have such a table?otfinfocan tell you or something like FontForge. If it does, you might be able to use the font for the maths in your document. Butunicode-mathis not the simple ride which the standard TeX fonts provide, so be prepared for bumps along the way. – cfr Feb 03 '18 at 04:29