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. 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:

$\int k{\frac {dy}{dx}}dx=k\int {\frac {dy}{dx}}dx.\quad$

Proof

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

$y=\int {\frac {dy}{dx}}dx.$

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

$ky=k\int {\frac {dy}{dx}}dx.\quad {\mbox{(1)}}$

Take the constant factor rule in differentiation:

${\frac {d\left(ky\right)}{dx}}=k{\frac {dy}{dx}}.$

Integrate with respect to x:

$ky=\int k{\frac {dy}{dx}}dx.\quad {\mbox{(2)}}$

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

$ky=k\int {\frac {dy}{dx}}dx$

$ky=\int k{\frac {dy}{dx}}dx.$

Therefore:

$\int k{\frac {dy}{dx}}dx=k\int {\frac {dy}{dx}}dx.\quad {\mbox{(3)}}$

Now make a new differentiable function:

$u={\frac {dy}{dx}}.$

Substitute in (3):

$\int kudx=k\int udx.$

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

$y=u.\,$

So:

$\int kydx=k\int ydx.$

This is the constant factor rule in integration.

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

$\int -ydx=-\int ydx.$