Integration of linear functions

[sgstudent]

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?

 

[Muphrid]

Yes, there's no meaningful difference there.

 

[sgstudent]

thanks!

 

[HallsofIvy]

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.

 

[Mentallic]

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.