Can someone tell me why this integral is invalid?

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Discussion Overview

The discussion revolves around the validity of a specific integral expression, focusing on the treatment of variables within the integral. Participants explore the conditions under which certain integrals can be considered valid or invalid, with an emphasis on the role of constants and functions in integration.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions the validity of the integral ∫ v Δp = pv + C, suggesting that v is being treated as a constant.
  • Another participant provides a comparison, stating that the integral ∫ k~dx = kx + C is valid because k is constant, while ∫ f(x)~dx = f(x) · x + C is not valid as f(x) is not constant.
  • Further contributions clarify that the expression ∫ y\,dx = yx + C holds true only if the derivative of y with respect to x is zero, implying that y must be a constant.
  • Participants reiterate that the condition for the validity of the integral is linked to whether the function being integrated is constant or variable.

Areas of Agreement / Disagreement

Participants generally agree on the principle that the validity of the integral depends on whether the variable in question is treated as a constant or not. However, the discussion does not reach a consensus on the specific integral in question.

Contextual Notes

The discussion highlights the importance of understanding the conditions under which integrals are valid, particularly in relation to the nature of the functions involved. There are unresolved aspects regarding the specific application of these principles to the integral presented.

Who May Find This Useful

Readers interested in mathematical reasoning, particularly in calculus and integration, may find this discussion relevant.

PhiPhenomenon
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Hey guys,

Can someone tell me why this integral isn't valid?

∫ v Δp = pv + C
 
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PhiPhenomenon said:
Hey guys,

Can someone tell me why this integral isn't valid?

∫ v Δp = pv + C

You're treating v as if it were a constant.

This is valid:

$$ \int k~dx = kx + C$$

This is not:
$$ \int f(x)~dx = f(x) \cdot x + C$$
 
Basically, \displaystyle \int y\,dx = yx + C \text{ iff }\frac{dy}{dx} = 0.
 
Millennial said:
Basically, \displaystyle \int y\,dx = yx + C \text{ iff }\frac{dy}{dx} = 0.
Which is the same as saying y is a constant.
 
Ah I see. Thanks guys.
 

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