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Prompt: Write a Python script for mathematical integration and format as LaTeX
Here is the Python implementation using **SciPy** and **SymPy** for numerical and symbolic integration:
```python
import sympy as sp
from scipy import integrate
Symbolic Integration with SymPy
x = sp.Symbol('x')
f = sp.exp(-x**2)
exact_result = sp.integrate(f, (x, 0, sp.oo))
print(f"Exact Result: {exact_result}")
Numerical Integration with SciPy
res, err = integrate.quad(lambda x: sp.exp(-x**2), 0, float('inf'))
print(f"Numerical Result: {res}")
```
Mathematical Formula
The Gaussian integral equation is represented as:
\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}Performance Comparison Matrix
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