SUMMARY
The integral of exp(-kx^2) over infinity is calculated using the substitution method. By letting u = x√k, the integral transforms to ∫_{-∞}^∞ e^{-u^2} du, which equals √π. Consequently, the final result for the integral ∫_{-∞}^∞ e^{-kx^2} dx is (1/√k) * √π, simplifying to √(π/k). This method is essential for evaluating Gaussian integrals in mathematical analysis.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with Gaussian integrals
- Knowledge of substitution methods in integration
- Basic algebraic manipulation skills
NEXT STEPS
- Study the properties of Gaussian integrals
- Learn about the substitution method in integral calculus
- Explore applications of integrals in probability theory
- Investigate the implications of k in Gaussian distributions
USEFUL FOR
Students and professionals in mathematics, physics, and engineering who require a solid understanding of Gaussian integrals and their applications in various fields.