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

[adamg]

if i integrate d/dx(x^2), should i include the constant of integration? thanks

 

[Crosson]

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.

 

[adamg]

so it doesn't matter that you know what the function was before differentiation?

 

[Crosson]

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).

 

[HallsofIvy]

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:

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