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:

[math]\displaystyle{ \int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx. \quad }[/math]

Proof

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

[math]\displaystyle{ 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]\displaystyle{ ky = k \int \frac{dy}{dx} dx. \quad \mbox{(1)} }[/math]

Take the constant factor rule in differentiation:

[math]\displaystyle{ \frac{d\left(ky\right)}{dx} = k \frac{dy}{dx}. }[/math]

Integrate with respect to x:

[math]\displaystyle{ ky = \int k \frac{dy}{dx} dx. \quad \mbox{(2)} }[/math]

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

[math]\displaystyle{ ky = k \int \frac{dy}{dx} dx }[/math]

[math]\displaystyle{ ky = \int k \frac{dy}{dx} dx. }[/math]

Therefore:

[math]\displaystyle{ \int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx. \quad \mbox{(3)} }[/math]

Now make a new differentiable function:

[math]\displaystyle{ u = \frac{dy}{dx}. }[/math]

Substitute in (3):

[math]\displaystyle{ \int ku dx = k \int u dx. }[/math]

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

[math]\displaystyle{ y = u. \, }[/math]

So:

[math]\displaystyle{ \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]\displaystyle{ \int -y dx = -\int y dx. }[/math]