Integration of linear functions

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Homework Help Overview

The discussion revolves around the integration of the linear function (2x + 5) and the implications of using different integration methods on the constant of integration. Participants explore the outcomes of integrating using the simplest rule versus a linear function method.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants examine the results of integrating (2x + 5) through different methods and question whether the constants of integration derived from each method are equivalent. Some express confusion regarding the treatment of arbitrary constants in integration.

Discussion Status

The conversation includes attempts to clarify the nature of the constant of integration and how it can vary based on the method used. Some participants suggest that the constants can be merged or treated as equivalent under certain conditions, while others provide examples to illustrate their points.

Contextual Notes

There is a repeated emphasis on the arbitrary nature of the constant of integration, and some participants reference specific conditions that could affect the value of this constant, such as initial conditions in differential equations.

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Homework Statement


If i integrate (2x+5) using the simplest rule, i get x2+5x+c, however if i use the linear function method, i would get (2x+5)^2/4 +c so simplifying it i get (4x^2+20x+25)/4 +c=x^2+5x+25/4+c. So does it mean when i use the two different methods my constant,c is different?


Homework Equations


linear integration rule

The Attempt at a Solution


So does it mean when i use the two different methods my constant,c is different?
 
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Yes, there's no meaningful difference there.
 
thanks!
 
sgstudent said:

Homework Statement


If i integrate (2x+5) using the simplest rule, i get x2+5x+c, however if i use the linear function method, i would get (2x+5)^2/4 +c so simplifying it i get (4x^2+20x+25)/4 +c=x^2+5x+25/4+c. So does it mean when i use the two different methods my constant,c is different?


Homework Equations


linear integration rule

The Attempt at a Solution


So does it mean when i use the two different methods my constant,c is different?
C is an arbitrary constant. It doesn't matter whether you use "c" or "25/4+ c" it still represents some arbitrary constant.

Suppose you were given that dy/dx= 2x+ 5 and that y(0)= 1. Clearly y is the integral of 2x+ 5 so, integrating the first way, you would get y(x)= x2+ 5x+ c and then the addtional condition, that y(0)= 1, becomes y(0)= 02+ 5(0)+ c= c= 1 so you answer is y(x)= x2+ 5x+ 1.

Integrating the second way, y(x)= x2+ 5x+ 25/4+ c so that y(0)= 02+ 5(0)+ 25/4+ c= 25/4+ x= 1 so that c= 1- 25/4= -21/4. But that means that y(x)= x2+ 5x+ 25/4- 21/4= x2+ 5x+ 4/4= x2+ 5x+ 1 just as before.
 
sgstudent said:

Homework Statement


If i integrate (2x+5) using the simplest rule, i get x2+5x+c, however if i use the linear function method, i would get (2x+5)^2/4 +c so simplifying it i get (4x^2+20x+25)/4 +c=x^2+5x+25/4+c. So does it mean when i use the two different methods my constant,c is different?


Homework Equations


linear integration rule

The Attempt at a Solution


So does it mean when i use the two different methods my constant,c is different?

If I ever use a method of integration that gives me some constant such as +1 but then I also have a +c added on the end, I just merge them together to make it +c2 or +k or if the teacher isn't strict / doesn't care I'll just get rid of the 1.
 

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