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:
