Why are C++ inline functions in the header?

There are two ways to look at it:

1. Inline functions are defined in the header because, in order to inline a function call, the compiler must be able to see the function body. For a naive compiler to do that, the function body must be in the same translation unit as the call. (A modern compiler can optimize across translation units, and so a function call may be inlined even though the function definition is in a separate translation unit, but these optimizations are expensive, aren't always enabled, and weren't always supported by the compiler)

2. functions defined in the header must be marked inline because otherwise, every translation unit which includes the header will contain a definition of the function, and the linker will complain about multiple definitions (a violation of the One Definition Rule). The inline keyword suppresses this, allowing multiple translation units to contain (identical) definitions.

The two explanations really boil down to the fact that the inline keyword doesn't exactly do what you'd expect.

A C++ compiler is free to apply the inlining optimization (replace a function call with the body of the called function, saving the call overhead) any time it likes, as long as it doesn't alter the observable behavior of the program.

The inline keyword makes it easier for the compiler to apply this optimization, by allowing the function definition to be visible in multiple translation units, but using the keyword doesn't mean the compiler has to inline the function, and not using the keyword doesn't forbid the compiler from inlining the function.

The definition of an inline function doesn't have to be in a header file but, because of the one definition rule (ODR) for inline functions, an identical definition for the function must exist in every translation unit that uses it.

The easiest way to achieve this is by putting the definition in a header file.

If you want to put the definition of a function in a single source file then you shouldn't declare it inline. A function not declared inline does not mean that the compiler cannot inline the function.

Whether you should declare a function inline or not is usually a choice that you should make based on which version of the one definition rules it makes most sense for you to follow; adding inline and then being restricted by the subsequent constraints makes little sense.