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March 25, 2017

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Given two problems:

Problem A: Tau = Integral(0 to v)
v/(q^2 - v^2) dv
q = constant

Problem B: Tau = Integral(V TO 0)
v/(q^2 - v^2) dv
q = constant

Why is it that, SUBSTITUTION RULE is used in problem A, and QUOTIENT RULE is used in problem B? The two problems are somewhat similar. The only different is their integrals. Problem A, the integration is from 0 to V and problem B, the integration is from V to 0. The answers are shown below. Please help!!

Answer (problem A):
Tau = ln(q^2/(q^2-v^2))

Answer (problem B):
Tau = ln((q^2-v^2)/q^2)

  • Calculus - ,

    One integral is the negative of the other, since the integrand is the same and only the direction of integration changes. If you think about the answers, and recognize that log a/b = - log b/a, you will see that all that is involved is a sign change there also. Either integration could have been solved by either method.

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