Mathematics

Mathematics

Problem

The value of ∫0∞e−a2x2dx\int_{0}^{\infty} e^{-a^{2}x^{2}} dx is equal to a. π2a\frac{\sqrt{\pi}}{2a} b. π2a\frac{\pi}{2a} c. π2a\frac{\pi}{\sqrt{2a}} d. None of these
Intuitive explanation

This is a classic Gaussian integral—one of the most important in calculus and physics. The key insight is that you can't solve it with standard antiderivatives, so instead you'll square the integral, convert to polar coordinates, and exploit the symmetry of the exponential function. The parameter aa just scales the width of the bell curve, which you can track through a substitution.

Intuitive explanation

This is a classic Gaussian integral—one of the most important in calculus and physics. The key insight is that you can't solve it with standard antiderivatives, so instead you'll square the integral, convert to polar coordinates, and exploit the symmetry of the exponential function. The parameter aa just scales the width of the bell curve, which you can track through a substitution.

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