3. Evaluate the following integrals:BJ(J+1)a) | BJ(J+1)(2J+1)eRTdJ

We have given ∫0∞BJ(J+1)(2J+1)e-BJ(J+1)kTdJput BJ(J+1)kT=Z⇒BJ(J+1)=kTZ…

Answered: 3. Evaluate the following integrals:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Evaluate the following integrals:
BJ(J+1)
a) | BJ(J+1)(2J+1)e
RT
dJ

Expert Solution

Step 1

$W e h a v e g i v e n \int_{0}^{\infty} B J (J + 1) (2 J + 1) e^{- \frac{B J (J + 1)}{k T}} d J p u t \frac{B J (J + 1)}{k T} = Z \Rightarrow B J (J + 1) = k T Z B (2 J + 1) d J = k T d Z \Rightarrow (2 J + 1) d J = \frac{k T}{B} d Z J = 0 \Rightarrow Z = 0 J = \infty \Rightarrow Z = \infty \int_{0}^{\infty} \frac{{(k T)}^{2}}{B} Z e^{- Z} d Z = \frac{{(k T)}^{2}}{B} \int_{0}^{\infty} Z e^{- Z} d Z = \frac{{(k T)}^{2}}{B} \int_{0}^{\infty} Z^{2 - 1} e^{- Z} d Z (τ (n) = \int_{0}^{\infty} Z^{n - 1} e^{- Z} d Z) = \frac{{(k T)}^{2}}{B} τ (2)$

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