Pure Mathematics is the branch of mathematics that deals only
with mathematics and how it works - it is the HOW of mathematics.
It is abstracted from the real world and provides the "tool box" of
mathematics; it includes things like calculus.
Applied mathematics is the branch of mathematics which applies
the techniques of Pure Mathematics to the real world - it is the
WHERE of mathematics; it includes things like mechanics.
Pure Mathematics teaches you HOW to integrate, Applied
mathematics teaches you WHERE to use integration.