Rules for integration

Integral_subs_rules.txt

Indefinite integrals:

1. If old is an integration variable, and new is a Symbol that is not present in expr, then expr.subs({old: new}), should xreplace old with new.

Example:

expr = Integral(f(x), x)
expr.subs(x, y) // should give
⌠        
⎮ f(y) dy
⌡        
since dx = dy

However
expr = Integral(f(y), x)
expr.subs(x, y) should NOT give
⌠        
⎮ f(y) dy
⌡        

2. If old is an integration variable and new is anything, then it should give a definite integral with new as the upper bound.

expr = Integral(f(x), x)
expr.subs(x, exp(x)) should give
 x        
ℯ         
⌠         
⎮  f(x) dx
⌡         
expr.subs(x, exp(y)) should give

 y        
ℯ         
⌠         
⎮  f(x) dx
⌡       


3. If old is a function of the integration variable and new is also a function of the integration variable.

expr = Integral(exp(x)*log(x), x)
expr.subs(expr.subs(log(x), exp(x)) // What should this return?


Definite integrals:

1. If old is an integration variable, and new is a Symbol that is not present in expr, then expr.subs({old: new}), should xreplace old with new. // Same as above

Example:

expr = Integral(f(x), (x, 1, 2))
expr.subs(x, y) // should give

2        
⌠        
⎮ f(y) dy
⌡        
1        

since dx = dy

However
expr = Integral(f(y), (x, 1, 2))
expr.subs(x, y) should NOT give
2        
⌠        
⎮ f(y) dy
⌡        
1

2. If old is an integration variable and new is anything, should it return a ValueError?


3. If old is a function of the integration variable and new is also a function of the integration variable, what should it do? Value Error?

According to me, any other case should return a ValueError/