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. 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:

${\displaystyle \int k{\frac {dy}{dx}}dx=k\int {\frac {dy}{dx}}dx.\quad }$

Video Constant factor rule in integration

Proof

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

${\displaystyle y=\int {\frac {dy}{dx}}dx.}$

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

${\displaystyle ky=k\int {\frac {dy}{dx}}dx.\quad {\mbox{(1)}}}$

Take the constant factor rule in differentiation:

${\displaystyle {\frac {d\left(ky\right)}{dx}}=k{\frac {dy}{dx}}.}$

Integrate with respect to x:

${\displaystyle ky=\int k{\frac {dy}{dx}}dx.\quad {\mbox{(2)}}}$

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

${\displaystyle ky=k\int {\frac {dy}{dx}}dx}$

${\displaystyle ky=\int k{\frac {dy}{dx}}dx.}$

Therefore:

${\displaystyle \int k{\frac {dy}{dx}}dx=k\int {\frac {dy}{dx}}dx.\quad {\mbox{(3)}}}$

Now make a new differentiable function:

${\displaystyle u={\frac {dy}{dx}}.}$

Substitute in (3):

${\displaystyle \int kudx=k\int udx.}$

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

${\displaystyle y=u.\,}$

So:

${\displaystyle \int kydx=k\int ydx.}$

This is the constant factor rule in integration.

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

${\displaystyle \int -ydx=-\int ydx.}$

Source of the article : Wikipedia