Gaussian integrals appear across probability, signal processing, and physics, making them foundational for both theoretical work and applied modeling. Mastering concrete Gaussian integral examples helps you estimate normalizing constants and evaluate likelihoods efficiently.
This article walks through standard, polar, and multivariable cases, emphasizing practical computation patterns rather than only abstract theory. The structured table and focused sections support rapid scanning while preserving mathematical clarity.
| Example Name | Integral Form | Key Technique | Result |
|---|---|---|---|
| Standard One-Dimensional | ∫-∞∞ e-x² dx | Square the integral and switch to polar coordinates | √π |
| Scaled One-Dimensional | ∫-∞∞ e-a x² dx, a > 0 | Substitution using √a x | √(π / a) |
| Gaussian with Linear Term | ∫-∞∞ e-a x² + b x dx | Complete the square in the exponent | √(π / a) eb² / (4a) |
| Two-Dimensional Orthogonal | ∫∫ e-(x² + y²) dx dy | Polar transformation (r, θ) | π |
Standard Gaussian Integral Computation
Begin with the benchmark integral ∫-∞∞ e-x² dx, where direct antiderivative in elementary functions does not exist. Define I as this integral, then compute I² as a double integral over the plane. Switching to polar coordinates replaces dx dy with r dr dθ and yields I² = π, so I = √π.
Scaling and Shifting Parameters
Effect of Positive Scaling
For ∫-∞∞ e-a x² dx with a > 0, substitute u = √a x so that dx = du / √a. The integral reduces to (1 / √a) ∫-∞∞ e-u² du, producing √(π / a). Larger a concentrates mass near zero and reduces total area accordingly.
Adding a Linear Term
To handle ∫-∞∞ e-a x² + b x dx, complete the square as -a(x − b / (2a))² + b² / (4a). Substitute y = x − b / (2a), after which the integral factorizes into eb² / (4a) ∫ e-a y² dy, giving √(π / a) eb² / (4a). This pattern is central to normal distribution moment calculations.
Multidimensional and Polar Techniques
Two-Dimensional Gaussian without Cross Terms
The integral ∫∫ e-(x² + y²) dx dy separates into the product of two identical one-dimensional Gaussian integrals, each equal to √π, so the double integral equals π. Alternatively, integrate in polar coordinates by letting x² + y² = r² and dx dy = r dr dθ over r in [0, ∞) and θ in [0, 2π], which directly delivers π as the result.
Radial Symmetry Extension
In higher dimensions, Gaussian integrals with radial exponent e-α r² are handled using spherical coordinates. The angular part contributes the surface area of the unit sphere, while the radial integral involves rn−1 e-α r², leading to formulas proportional to πn/2 / αn/2. These results underpin Gaussian measures in statistics and physics.
Computational Tips and Error Control
When implementing Gaussian integral approximations numerically, prefer stable parameterizations such as scaling before discretization. Truncation bounds should reflect decay speed: using ±4√(1/a) or ±4√(1/α) captures most probability mass for well-conditioned problems. Adaptive quadrature and change-of-variable techniques further improve accuracy without sacrificing interpretability.
Key Takeaways for Gaussian Integral Applications
- Use polar coordinates to evaluate the basic square-integral I² and derive √π.
- Apply scaling substitution to reduce general ∫ e-a x² dx to the standard √(π / a) form.
- Complete the square for linear-exponent terms, producing mean-shift factors eb² / (4a).
- Recognize separability and radial symmetry to extend results to multiple dimensions.
- Truncate infinite domains only after verifying that tails contribute negligibly to your tolerance.
FAQ
Reader questions
How do I compute the Gaussian integral with a negative quadratic coefficient if a is positive?
The integral converges only when the quadratic coefficient in the exponent is negative for the variable squared, meaning a > 0 in e -a x² . For e +a x² with a > 0, the integral diverges over infinite limits, so positive definiteness in the exponent is required for finite results.
Can I use the same completing-the-square trick when integrating over a finite interval?
Yes, you can complete the square and express the integral in terms of the error function erf, since ∫ e -a(x − μ)² dx over finite bounds maps to scaled erf values at the endpoints. The infinite-interval closed forms become approximations unless the bounds are wide enough that tail contributions are negligible.
What changes when the Gaussian has a covariance structure in multiple dimensions?
With a quadratic form x T Σ −1 x where Σ is positive definite, the multidimensional Gaussian integral over R n yields (2π) n/2 |Σ| 1/2 . The determinant captures volume scaling, and symmetry of Σ ensures the integral remains finite and tractable via diagonalization or Cholesky methods.
How does shifting the mean affect the integral value if I integrate over the real line?
Shifting by a linear term b x corresponds to a mean shift in the associated normal density up to proportionality. After completing the square, the integral over the real line retains the same Gaussian width parameter a but multiplies by e b² / (4a) , so the total area depends on both the scale a and the shift magnitude b.