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| {{Unreferenced|date=August 2010}}
| | Hello!<br><br>Ok, i'll start by saying my name - Lindsey Laforge. Connecticut is where my property is but I'm going to have to maneuver in one year or a couple of them. Her husband doesn't like it the way she does but what she really loves doing is astrology but she's been taking on new things lately. Hiring has been her [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=regular+job&btnI=lucky regular job] for a time but soon she'll be on her. You can always find his website here: http://www.pinterest.com/seodress/granite-bay-real-estate-expert-realtor-in-all-thin/ |
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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:
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| <math>\int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx. \quad </math>
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| == Proof == | |
| Start by noticing that, from the definition of [[integral|integration]] as the [[Inverse function|inverse]] process of [[derivative|differentiation]]:
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| :<math>y = \int \frac{dy}{dx} dx.</math>
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| Now [[product (mathematics)|multiply]] both sides by a [[Coefficient|constant]] ''k''. Since ''k'' is a constant it is [[not dependent on]] ''x'':
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| :<math>ky = k \int \frac{dy}{dx} dx. \quad \mbox{(1)}</math>
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| Take the [[constant factor rule in differentiation]]:
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| :<math>\frac{d\left(ky\right)}{dx} = k \frac{dy}{dx}.</math>
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| [[integral|Integrate]] with respect to ''x'':
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| :<math>ky = \int k \frac{dy}{dx} dx. \quad \mbox{(2)}</math> | |
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| Now from (1) and (2) we have:
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| :<math>ky = k \int \frac{dy}{dx} dx</math>
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| :<math>ky = \int k \frac{dy}{dx} dx.</math>
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| Therefore:
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| :<math>\int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx. \quad \mbox{(3)}</math>
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| Now make a new differentiable [[function (mathematics)|function]]:
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| :<math>u = \frac{dy}{dx}.</math>
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| [[Substitution property of equality|Substitute]] in (3):
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| :<math>\int ku dx = k \int u dx.</math>
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| Now we can re-substitute ''y'' for something different from what it was originally:
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| :<math>y = u. \,</math>
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| So:
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| :<math>\int ky dx = k \int y dx.</math>
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| This is the constant factor rule in integration.
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| A [[special case]] of this, with ''k''=-1, yields:
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| :<math>\int -y dx = -\int y dx.</math>
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| {{DEFAULTSORT:Constant Factor Rule In Integration}}
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| [[Category:Integral calculus]]
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| {{mathanalysis-stub}}
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Hello!
Ok, i'll start by saying my name - Lindsey Laforge. Connecticut is where my property is but I'm going to have to maneuver in one year or a couple of them. Her husband doesn't like it the way she does but what she really loves doing is astrology but she's been taking on new things lately. Hiring has been her regular job for a time but soon she'll be on her. You can always find his website here: http://www.pinterest.com/seodress/granite-bay-real-estate-expert-realtor-in-all-thin/