pythonsympysymbolic-math
sympy integration with cosine function under a square root
I am trying to solve the integration
integrate( sqrt(1 + cos(2 * x)), (x, 0, pi) )
Clearly, through pen and paper this is not hard, and the result is:
But when doing this through Sympy, something does not seem correct.
I tried the sympy codes as below.
from sympy import *
x = symbols("x", real=True)
integrate(sqrt(1 + cos(2 * x)), (x, 0, pi)).doit()
It then gives me a ValueError saying something in the complex domain not defined. But I've already defined the symbol x as a variable in the real domain.
Here is the full error:
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
Cell In[7], line 4
1 from sympy import *
3 x = symbols("x", real=True)
----> 4 integrate(sqrt(1 + cos(2 * x)), (x, 0, pi)).doit()
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:1567, in integrate(meijerg, conds, risch, heurisch, manual, *args, **kwargs)
1564 integral = Integral(*args, **kwargs)
1566 if isinstance(integral, Integral):
-> 1567 return integral.doit(**doit_flags)
1568 else:
1569 new_args = [a.doit(**doit_flags) if isinstance(a, Integral) else a
1570 for a in integral.args]
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:499, in Integral.doit(self, **hints)
497 if reps:
498 undo = {v: k for k, v in reps.items()}
--> 499 did = self.xreplace(reps).doit(**hints)
500 if isinstance(did, tuple): # when separate=True
501 did = tuple([i.xreplace(undo) for i in did])
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:710, in Integral.doit(self, **hints)
707 uneval = Add(*[eval_factored(f, x, a, b)
708 for f in integrals])
709 try:
--> 710 evalued = Add(*others)._eval_interval(x, a, b)
711 evalued_pw = piecewise_fold(Add(*piecewises))._eval_interval(x, a, b)
712 function = uneval + evalued + evalued_pw
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\core\expr.py:956, in Expr._eval_interval(self, x, a, b)
953 domain = Interval(b, a)
954 # check the singularities of self within the interval
955 # if singularities is a ConditionSet (not iterable), catch the exception and pass
--> 956 singularities = solveset(self.cancel().as_numer_denom()[1], x,
957 domain=domain)
958 for logterm in self.atoms(log):
959 singularities = singularities | solveset(logterm.args[0], x,
960 domain=domain)
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2252, in solveset(f, symbol, domain)
2250 if symbol not in _rc:
2251 x = _rc[0] if domain.is_subset(S.Reals) else _rc[1]
-> 2252 rv = solveset(f.xreplace({symbol: x}), x, domain)
2253 # try to use the original symbol if possible
2254 try:
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2276, in solveset(f, symbol, domain)
2273 f = f.xreplace({d: e})
2274 f = piecewise_fold(f)
-> 2276 return _solveset(f, symbol, domain, _check=True)
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:1060, in _solveset(f, symbol, domain, _check)
1057 result = Union(*[solver(m, symbol) for m in f.args])
1058 elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
1059 _is_function_class_equation(HyperbolicFunction, f, symbol):
-> 1060 result = _solve_trig(f, symbol, domain)
1061 elif isinstance(f, arg):
1062 a = f.args[0]
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:612, in _solve_trig(f, symbol, domain)
610 sol = None
611 try:
--> 612 sol = _solve_trig1(f, symbol, domain)
613 except _SolveTrig1Error:
614 try:
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:688, in _solve_trig1(f, symbol, domain)
685 if g.has(x) or h.has(x):
686 raise _SolveTrig1Error("change of variable not possible")
--> 688 solns = solveset_complex(g, y) - solveset_complex(h, y)
689 if isinstance(solns, ConditionSet):
690 raise _SolveTrig1Error("polynomial has ConditionSet solution")
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2284, in solveset_complex(f, symbol)
2283 def solveset_complex(f, symbol):
-> 2284 return solveset(f, symbol, S.Complexes)
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2252, in solveset(f, symbol, domain)
2250 if symbol not in _rc:
2251 x = _rc[0] if domain.is_subset(S.Reals) else _rc[1]
-> 2252 rv = solveset(f.xreplace({symbol: x}), x, domain)
2253 # try to use the original symbol if possible
2254 try:
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2276, in solveset(f, symbol, domain)
2273 f = f.xreplace({d: e})
2274 f = piecewise_fold(f)
-> 2276 return _solveset(f, symbol, domain, _check=True)
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:1110, in _solveset(f, symbol, domain, _check)
1106 result += _solve_radical(equation, u,
1107 symbol,
1108 solver)
1109 elif equation.has(Abs):
-> 1110 result += _solve_abs(f, symbol, domain)
1111 else:
1112 result_rational = _solve_as_rational(equation, symbol, domain)
File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:918, in _solve_abs(f, symbol, domain)
916 """ Helper function to solve equation involving absolute value function """
917 if not domain.is_subset(S.Reals):
--> 918 raise ValueError(filldedent('''
919 Absolute values cannot be inverted in the
920 complex domain.'''))
921 p, q, r = Wild('p'), Wild('q'), Wild('r')
922 pattern_match = f.match(p*Abs(q) + r) or {}
ValueError:
Absolute values cannot be inverted in the complex domain.
How do I properly integrate this using Sympy?
Solution
Adding a simplification in there will produce the correct result, but I'm not sure why it is having an issue in the first place.
integrate(sqrt(1+cos(2*x)).simplify(), (x, 0, pi)) # 2*sqrt(2)