I don't understand why sympy can't calculate very simple integral from 0 to t. How to solve this problem?
import sympy
from sympy import sin , cos , sqrt, asinh , log
t= sympy.symbols('t')
x = 't^2'
y = 't^3'
x_derivatives = sympy.diff(x , t)
y_derivatives = sympy.diff(y , t)
expression = x_derivatives**2 + y_derivatives**2
print(expression)
fi_t = sympy.integrate(sympy.sqrt(expression), (t,0,t))
print(fi_t)
Result:
9*t**4 + 4*t**2
Integral(sqrt(9*t**4 + 4*t**2), (t, 0, t))
It seems that sympy's integrate function struggles with this integral but we can help it along by showing what substitution to use:
In [46]: fi_t = Integral(sqrt(9*t**4 + 4*t**2), (t, 0, t))
In [47]: fi_t
Out[47]:
t
⌠
⎮ _____________
⎮ ╱ 4 2
⎮ ╲╱ 9⋅t + 4⋅t dt
⌡
0
In [48]: z = Symbol('z', positive=True)
In [49]: fi_t.transform(t, sqrt(z))
Out[49]:
2
t
⌠
⎮ ____________
⎮ ╱ 2
⎮ ╲╱ 9⋅z + 4⋅z
⎮ ─────────────── dz
⎮ 2⋅√z
⌡
0
In [50]: factor_terms(fi_t.transform(t, sqrt(z)))
Out[50]:
2
t
⌠
⎮ _________
⎮ ╲╱ 9⋅z + 4 dz
⌡
0
─────────────────
2
In [51]: factor_terms(fi_t.transform(t, sqrt(z))).doit()
Out[51]:
3/2
⎛ 2 ⎞
⎝9⋅t + 4⎠ 8
───────────── - ──
27 27
I re-run your code in the in-build sympy shell. Changing t in the upper bound of the integral to 1 fixes the code.
import sympy
from sympy import sin , cos , sqrt, asinh , log
t= sympy.symbols('t')
x = 't^2'
y = 't^3'
x_derivatives = sympy.diff(x , t)
y_derivatives = sympy.diff(y , t)
expression = x_derivatives**2 + y_derivatives**2
print(expression)
def fi_t(x):
return sympy.integrate(sympy.sqrt(expression), (t,0,x))
print(fi_t(4))
print(fi_t(-4))
sympy.integrate(sympy.sqrt(expression), (t,0,t)),tis a variable and an upper bound, which is impossible. - Kate Melnykovatin(t,0,t)must be a constant, no? - DYZ(t, -4, 4). - DYZ