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.

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