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.
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>