| (0.5*x + sin(x)) dx
|
/
a
Using integrate, we can then derive the antiderivative of the integration function:
In [ 101 ]: int_func = sy.integrate(sy.sin(x) + 0.5 * x, x)
In [ 102 ]: print sy.pretty(int_func)
Out[102]: 2
0.25*x - cos(x)
Equipped with the antiderivative, the numerical evaluation of the integral is only three
steps away. To numerically evaluate a SymPy expression, replace the respective symbol
with the numerical value using the method subs and call the method evalf on the new
expression:
In [ 103 ]: Fb = int_func.subs(x, 9.5).evalf()
Fa = int_func.subs(x, 0.5).evalf()
The difference between Fb and Fa then yields the exact integral value:
In [ 104 ]: Fb - Fa # exact value of integral
Out[104]: 24.3747547180867
The integral can also be solved symbolically with the symbolic integration limits:
In [ 105 ]: int_func_limits = sy.integrate(sy.sin(x) + 0.5 * x, (x, a, b))
print sy.pretty(int_func_limits)
Out[105]: 2 2
- 0.25*a + 0.25*b + cos(a) - cos(b)
As before, numerical substitution — this time using a dict object for multiple
substitutions — and evaluation then yields the integral value:
In [ 106 ]: int_func_limits.subs({a : 0.5, b : 9.5}).evalf()
Out[106]: 24.3747547180868
Finally, providing quantified integration limits yields the exact value in a single step:
In [ 107 ]: sy.integrate(sy.sin(x) + 0.5 * x, (x, 0.5, 9.5))
Out[107]: 24.3747547180867
Differentiation
The derivative of the antiderivative shall yield in general the original function. Let us
check this by applying the diff function to the symbolic antiderivative from before:
In [ 108 ]: int_func.diff()
Out[108]: 0.5*x + sin(x)
As with the integration example, we want to use differentiation now to derive the exact
solution of the convex minimization problem we looked at earlier. To this end, we define
the respective function symbolically as follows:
In [ 109 ]: f = (sy.sin(x) + 0.05 * x ** 2
+ sy.sin(y) + 0.05 * y ** 2 )
For the minimization, we need the two partial derivatives with respect to both variables, x
and y:
In [ 110 ]: del_x = sy.diff(f, x)
del_x
Out[110]: 0.1*x + cos(x)
In [ 111 ]: del_y = sy.diff(f, y)
del_y
