Constant factor rule in integration

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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. It states that a constant factor within an integrand can be separated from the integrand and instead multiplied by the integral. For example, where k is a constant:

Proof[edit]

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:

Substitute 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: