Constant factor rule in integration

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

<math>y = \int \frac{dy}{dx} dx</math>

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

<math>ky = k \int \frac{dy}{dx} dx \quad \mbox{(1)}</math>

Take the constant factor rule in differentiation:

<math>\frac{d\left(ky\right)}{dx} = k \frac{dy}{dx}</math>

Integrate with respect to x:

<math>ky = \int k \frac{dy}{dx} dx \quad \mbox{(2)}</math>

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

<math>ky = k \int \frac{dy}{dx} dx</math>

<math>ky = \int k \frac{dy}{dx} dx</math>

Therefore:

<math>\int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx \quad \mbox{(3)}</math>

Now make a new differentiable function:

<math>u = \frac{dy}{dx}</math>

Subsitute[?] in (3):

<math>\int ku dx = k \int u dx</math>

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

<math>y = u</math>

So:

<math>\int ky dx = k \int y dx</math>

This is the constant factor rule in integration.

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

<math>\int -y dx = -\int y dx</math>