How to calculate definite integral in Python

Definite integrals are fundamental in mathematics, physics, and engineering. Python offers multiple libraries for exact and numerical integration. This guide covers four primary methods: symbolic integration with SymPy, numerical quadrature with SciPy, arbitrary-precision integration with mpmath, and discrete approximation with NumPy.

1. Symbolic Integration with SymPy

Use sympy.integrate for exact antiderivatives and definite integrals.

from sympy import symbols, integrate, sin

x = symbols('x')
expr = sin(x)**2

# Definite integral from 0 to pi
result = integrate(expr, (x, 0, 3.141592653589793))
print(result)  # pi/2
  

Tip: SymPy handles a wide range of symbolic integrals but may be slow for complex expressions.

2. Numerical Quadrature with SciPy

scipy.integrate.quad uses adaptive algorithms for high-accuracy numerical integration.

import numpy as np
from scipy.integrate import quad

# Integrand
def f(x):
    return np.sin(x)**2

# Compute integral
result, error = quad(f, 0, np.pi)
print(f"Result: {result:.6f}, Error estimate: {error:.2e}")  # ~1.570796, ~0
  

Note: quad returns both value and error estimate. For oscillatory or singular functions, consider quadpack options.

3. Arbitrary-Precision Integration with mpmath

mpmath.quad supports arbitrary precision and complex integrals.

import mpmath as mp

mp.mp.dps = 50  # 50 digits precision

def g(x):
    return mp.exp(-x**2)

# Integrate from 0 to infinity
result = mp.quad(g, [0, mp.inf])
print(result)  # ~0.886226925...
  

Tip: Use mp.quad with piecewise intervals ([a,b]) for infinite limits.

4. Discrete Approximation with NumPy

Approximate integrals using the trapezoidal rule with numpy.trapz.

import numpy as np

x = np.linspace(0, np.pi, 100000)
y = np.sin(x)**2

# Trapezoidal rule
result = np.trapz(y, x)
print(result)  # ~1.570796
  

Note: Increase sample points for higher accuracy. For Simpson’s rule, use scipy.integrate.simpson.

Comparison of Methods

Method	Use Case	Accuracy	Performance
SymPy	Exact symbolic integrals	Exact	Slow for complex
SciPy quad	General numerical	Machine precision	Fast
mpmath	High-precision or complex	Arbitrary	Moderate
NumPy trapz	Quick approximations	Depends on sampling	Very fast

Summary Checklist

1. Use SymPy for closed-form results.
2. Use SciPy quad for reliable numerical integration.
3. Use mpmath for high-precision or complex path integrals.
4. Use NumPy trapz for quick, large-scale approximations.