Constant factor rule in integration

The constant factor rule in integration is a dual of the constant factor rule in differentiation, and is a consequence of the linearity of integration

Start by noticing that, from the definition of integration as the inverse process of differentiation:

Now multiply both sides by a constant k. Since k is a constant it is not dependent on x:

Take the constant factor rule in differentiation:

Integrate with respect to x:

Now from (1) and (2) we have:

Therefore:

Now make a new differentiable function:

Subsitute in (3):

Now we can re-substitute y for something different from what it was originally:

So:

This is the constant factor rule in integration.

A special case of this, with k=-1, yields: