Hello, @Nasser. This behavior is related to the code of the `integrate`

function itself. If you execute `integrate??`

in your Sage session, you will be able to see the docstring of the subroutine, and its source code. We are interested in the source code right now:

```
if hasattr(x, 'integral'):
return x.integral(*args, **kwds)
else:
from sage.symbolic.ring import SR
return SR(x).integral(*args, **kwds)
```

This basically verifies if the object you passed as argument (`x`

in this code) has the `integral`

attribute, which is nothing more than a subroutine that specifies how `x`

should be integrated. If `x`

doesn't have such an attribute defined, then Sage converts it into an element of the Symbolic Ring (`SR`

).

In the case of your example, the string `"x"`

does not have an `integral`

attribute (since it's a string), so Sage calls `SR('x')`

, which converts your string into the identity function, i.e. $x\mapsto x$, whose integral is $1/2x^2$.

As particular examples, you can try:

```
f(x) = SR("x")
f(2)
```

This will return `2`

, as expected, showing you that you have actually define a function from its string representation. Now try:

```
(f(x)).integral(x)
```

which is equivalent to `integrate(f(x), x)`

or `integrate("x", x)`

of `(SR("x")).integral(x)`

. Finally, try:

```
f(x) = SR("sin(x)")
f(pi)
(f(x)).integral(x)
integrate("sin(x)", x)
```

**EDIT**
Based on the answer to this question, I must point out that the use of strings as functions, by means of the Symbolic Ring `SR`

, is discourage and should be avoided, because it has unpleasant consequences. This kind of use of strings seems not to be documented for that reason.

I hope this helps!