Integrate d/dx(x^2): Include Constant?

  • Context: High School 
  • Thread starter Thread starter adamg
  • Start date Start date
  • Tags Tags
    Integrate
Click For Summary

Discussion Overview

The discussion revolves around the integration of the derivative of the function \(x^2\) and whether a constant of integration should be included in the result. It touches on concepts of differentiation and integration, as well as the implications of including or excluding constants in various contexts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asks if a constant of integration should be included when integrating \(d/dx(x^2)\).
  • Another participant asserts that a constant of integration should be included when integrating both sides of an equation with respect to \(x\).
  • A question is raised about the relevance of knowing the original function before differentiation in the context of integration.
  • One participant expresses concern that excluding the constant could lead to inconsistent results depending on the order of operations.
  • Another participant clarifies that the inclusion of a constant depends on the type of integral being performed, distinguishing between indefinite integrals and definite integrals.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of including a constant of integration, with some asserting it is essential while others question its relevance based on the context of the integral being performed. The discussion remains unresolved regarding the implications of including or excluding the constant.

Contextual Notes

Participants highlight the dependence on the type of integral being considered, indicating that the discussion may be limited by assumptions about the context in which integration is applied.

adamg
Messages
48
Reaction score
0
if i integrate d/dx(x^2), should i include the constant of integration? thanks
 
Physics news on Phys.org
If you are working on an equation, then presumably you are integrating both sides with respect to x in which case you will have a constant of Integration (arbitrarily) on either side.

The short answer is yes, in all cases.
 
so it doesn't matter that you know what the function was before differentiation?
 
I am tempted to say that it wouldn't matter, but that would lead to inconsistent results (i.e. a different answer depending on the chosen order of operations).
 
What in the world do you mean? If you start with a function f(x), differentiate it, then integrate that, whether you get the original function, that function plus an unknown constant, or that function plus a specific number depends on exactly what type of "integral" you are doing:

[itex]\int f(x)dx[/itex], the indefinite integral should have an unknown constant added because it means ALL functions whose derivative is f(x) but [itex]\int_a^xf(t)dt[/itex] would not and the value will depend upon the choice of a.
 

Similar threads

High School Integration by Parts, an introduction I get confused with
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Non solvable integral? (dx/dt)^2 dt
  • · Replies 3 ·
Replies
3
Views
4K
High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
4K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
3K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
4K
High School Arc Length for Hyperbolic Sin
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
5K
Undergrad Another Integration Formula
  • · Replies 6 ·
Replies
6
Views
3K