Inconsistent results from "int" function

Hi, everyone. For the same question, the "int" function provided me with different results.

syms x y

int(x*y/24+y^2/48,x)

int((y*(2*x + y))/48,x)

The second line gave me "(x*y*(x + y))/48" which was what I originally wanted,

but the third line gave me "'(y*(2*x + y)^2)/192" which is wrong.

Can anyone explain the reason?

Thanks for your time and help in advance!

Answer 1

[Alan Stevens]

You can express '(y*(2*x + y)^2)/192 as (x*y*(x + y))/48+y^3/198

As far as the integral is concerned y is a constant, so the two results are the same to within a constant, which is all you can guarantee from an integral with no specific limits.

[YoungJoo Lee]

Dear Alan,

Thanks for your kind answer. I am also aware that the problem is about an indefinite integral, and the result may get involved with a constant. However, I think there is no mathematically-valid reason that the constant can be specified as y^3/192. If this is how the "int" function works, it seems the function cannot be used for indefinite integral problems...

[Torsten]

However, I think there is no mathematically-valid reason that the constant can be specified as y^3/192.

There is a mathematically-valid reason that each function f that only depends on y can be specified as the constant of integration. The reason is d/dx(f(y)) = 0.

MATLAB does not perform an "evaluation of the most beautiful antiderivative".

In this case, you can use

syms x y
int(x*y/24+y^2/48,x,0,x)

ans = 

int((y*(2*x + y))/48,x,0,x)

ans = 

But in other examples, the lower limit might be different from 0 to get a "nice" antiderivative.

[YoungJoo Lee]

Dear Torsten,

I appreciate you on your detailed explanation. I agree with you that "any" function f that only depends on y can be the constant of the integration in my problem. However, I am still doubting whether it is reasonable to provide only one specific function for y (e.g., y^3/192 in my case) as output, as if it were the only and unique answer...

I believe, for my problem, providing something like "(x*y*(x + y))/48+C (where C is a constant)" as output would be more appropriate.

Anyway, I got your point and also thank you for sharing your tip.

[Torsten]

I believe, for my problem, providing something like "(x*y*(x + y))/48+C (where C is a constant)" as output would be more appropriate.

"(x*y*(x + y))/48+C(y) (where C is a function solely depending on y)"

[YoungJoo Lee]

Oh, sure. Thanks for the correction!