State why each integral is improper, and determine whether or not it is converge

Question

State why each integral is improper, and determine whether or not it is convergent. If the improper integral is convergent, find its value. Show work.

S (from 1 to 2) x/(x-1)^1/2 dx

Explanation / Answer

LCM of 2 and 3 is 6. So take x = t^6 => dx = 6t^5 dt => ? dx /[x^(1/2) - x^(1/3)] = 6 ? t^5 dt / (t^3 - t^2) = 6 ? t^3 dt (t - 1) = 6 ? (t^3 - 1 + 1) dt /(t - 1) = 6 ? [(t - 1)(t^2 + t + 1) + 1] dt / (t - 1) = 6 ? [t^2 + t + 1 + 1/(t-1)] dt = 6 [t^3/3 + t^2/2 + t + ln l t - 1 l] + c = 2t^3 + 3t^2 + 6t + 6 ln l t - 1 l + c = 2x^(1/2) + 3x^(1/3) + 6x^(1/6) + 6 ln l x^(1/6) - 1 l + c

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