MATLAB: Symbolic Integration

symbolic integrationSymbolic Math Toolbox

Hi everybody,

I have a little question about 'int' function. I have this code:

>> syms x y  
>> int(x+y,x)

so I want to know why is this the result?:

ans =   
(x + y)^2/2

instead of:

x^2/2+x*y

I'm a little confused about it, I hope you can help me understand.

Best Answer

Both answers are correct.

d/dx[ (x+y)^2/2 ] = d/dx[ x^2/2 + x*y + y^2/2 ] = x + y

d/dx[ x^2/2 + x*y ] = x + y

The ambiguity lies in the constant of integration. In the first case the constant of integration ends up being y^2/2.

If you really want to enforce that the constant of integration is zero then do:

F = int(x+y,x,0,x)
expand(F)

Related Solutions

MATLAB: Int(exp,var) gives different results for same symbolic expression

Both solutions are equivalent and differ by a constant factor. The ambiguity can be resolved by using a condition (like the value of the function at initial point).

syms x y a
E1 = a*(x+y);
E2 = a*x + a*y;
y1 = expand(int(E1,x))
y2 = expand(int(E2,x))

Result

y1 =
(a*x^2)/2 + a*x*y + (a*y^2)/2
y2 =
(a*x^2)/2 + a*y*x

differes by a term containing 'y'. If you take the derivative of both functions w.r.t. 'x', you will get the original function back.

MATLAB: Does x(x+y) dy give a x^3 component

Both answers are correct.

d/dy[ x*(x+y)^2/2 ] = d/dy[ (x^3)/2 + x^2*y + y^2*x/2 ] = x*(x + y)

d/dy[ x*y*(2*x+y))/2 ] = x*(x + y)

The ambiguity lies in the constant of integration. In the first case the constant of integration ends up being (x^3)/2.

If you really want to enforce that the constant of integration is zero then do:

int(gx,y,0,y)

In other words

int(x*(x+y),y,0,y)

Best Answer

Related Solutions

MATLAB: Int(exp,var) gives different results for same symbolic expression

MATLAB: Does x(x+y) dy give a x^3 component

Related Question