An Introduction to Thermal Physics · Problem B.3

Systems of Interacting Particles — Daniel Schroeder

Physics — Statistical Mechanics · gaussian-integrals-odd

Problem

The integral of xne−ax2x^n e^{-ax^2} is easier to evaluate when nn is odd. (a) Evaluate ∫−∞∞xe−ax2 dx\displaystyle\int_{-\infty}^{\infty} x e^{-ax^2}\, dx. (No computation allowed!) (b) Evaluate the indefinite integral (i.e., the antiderivative) of xe−ax2x e^{-ax^2}, using a simple substitution. (c) Evaluate ∫0∞xe−ax2 dx\displaystyle\int_0^{\infty} x e^{-ax^2}\, dx. (d) Differentiate the previous result to evaluate ∫0∞x3e−ax2 dx\displaystyle\int_0^{\infty} x^3 e^{-ax^2}\, dx.

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