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Constant Factor Rule In IntegrationThe 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: -
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