What are some techniques for integration using SymPy?

What are some techniques for integration using SymPy?

SymPy offers several techniques for integration, including symbolic integration, numerical integration, and integration by parts. Would you like to see examples of each?

Yes, please show me an example of symbolic integration.

Sure! For symbolic integration, you can use the integrate function. For example, to integrate sin(x), you would write:

from sympy import symbols, integrate, sin
x = symbols('x')
integral = integrate(sin(x), x)
print(integral)

This will return -cos(x).

What about numerical integration?

For numerical integration, you can use the quad function from the scipy.integrate module. Here's an example:

from scipy.integrate import quad
import numpy as np

result, error = quad(np.sin, 0, np.pi)
print(result)

This will give you the numerical value of the integral of sin(x) from 0 to π.

Can you explain integration by parts?

Certainly! Integration by parts is based on the formula:

[ \int u , dv = uv - \int v , du ]

In SymPy, you can implement it like this:

from sympy import symbols, integrate, exp
x = symbols('x')
u = x
v = exp(x)
du = nu.diff(x)
v = integrate(v, x)

integral = nu * v - integrate(v * du, x)
print(integral)

This will compute the integral using the integration by parts technique.