Update app.py

app.py

@@ -281,19 +281,38 @@ NOTE: For eigenvalue problems, use 'lam = Symbol('lam')' instead of importing fr
 
    - **Important Note**: Always test limits symbolically first. If SymPy produces unexpected results, simplify the expression or expand it (e.g., using `series`) before re-evaluating the limit.
 
-   8. If calculating integrals, write the SymPy code to compute the original integral directly with respect to the original variable of integration,
-      not any integral that might have been rewritten with a variable substitution. In addition to seeking an analytic solution, also obtain a numerical solution using two methods
-      (i) SymPy's integrate function; and
-      (ii) using mpmath, example:
-      ```python
-            import mpmath 
-            f = lambda x: x**2 / ((x**4+1)*mpmath.sqrt(x**2+1))
-            mpmath.quad(f, [0, mpmath.inf])
-      ```
+8. If calculating integrals, do so directly with respect to the original expression and original variable of integration,
+      not any integral that might have been rewritten with a variable substitution
+   
+   For indefinite integrals:
+   ```python
+   # Just compute and display the symbolic result
+   x = Symbol('x')
+   expr = x**2
+   indefinite_integral = integrate(expr, x)
+   print("Indefinite integral:")
+   print(indefinite_integral)
+   ```
+
+   For numerical verification, always use definite integrals:
+   ```python
+   # Use specific bounds and verify numerically
+   definite_integral = integrate(expr, (x, 0, 1))
+   print("Definite integral value:")
+   print(float(definite_integral.evalf()))
+   
+   # Include mpmath verification:
+   import mpmath
+   f = lambda x: x**2
+   mpmath_result = mpmath.quad(f, [0, 1])
+   print("mpmath verification:", float(mpmath_result))
+   ```
 
    9. If using SymPy to evalute an infinite sum, attempt using the infinite sum, and in addition report what the sum of the first 100 terms is as a sanity check.
 
    10. Do not use SciPy.
+
+   11. Always use this template when working with systems of equations to handle potential linear dependence correctly. 
       
 ```python
 from sympy import Matrix, symbols, solve
@@ -354,7 +373,7 @@ if result == "infinite":
         else:
             print(f"{var} = t")  # use different parameter names for multiple free variables
 ```
-Always use this template when working with systems of equations to handle potential linear dependence correctly. """
+"""
 
 def load_proof_repository():
     """Load the proof repository from the repository file"""