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 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}.
Subsitute in (3):
\int ku dx = k \int u dx.
Now we can re-substitute y for something different from what it was originally:
y = u.
So:
\int ky dx = k \int y dx.
This is the constant factor rule in integration. A special case of this, with k=-1, yields:
\int -y dx = -\int y dx.

 

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