Berechne die folgenden uneigentlichen Integrale.

  1. ´int_0^1 dx/sqrt(x)´
  2. ´int_1^2 dx/(x - 1)^2´
  3. ´int_1^oo dx/x^2´
Approach

"Subtask 1\n\n´int 1/sqrt(x) dx´\n\nThe integral of ´1/sqrt(x)´ is ´2 sqrt(x)´:\n´= 2 sqrt(x) + "constant"´\n\n´int_0^1 dx/sqrt(x) = (2 sqrt(1) + "constant") - (2 sqrt(0) + "constant") = 2´\n\n\nSubtask 2\n\n´int 1/(x-1)^2 dx´\n\nFor the integrand ´1/(x-1)^2´, substitute ´u = x-1´ and ´du = dx´:\n´= int 1/u^2 du´\n\nThe integral of ´1/u^2´ is ´-1/u´:\n´= -1/u + "constant"´\n\nSubstitute back for ´u = x-1´:\n´= 1/(1-x) + "constant"´\n\n\n´int_1^2 dx/(x - 1)^2 =´ The Integral does not converge!\n\n\nSubtask 3\n\n´int 1/x^2 dx = -1/x + "constant"´\n\n´int_1^oo dx/x^2 = 1´"


Solution
    1. ´2´
    2. The Integral does not converge!
    3. ´1´
  • URL:
  • Language: Deutsch
  • Subjects: math
  • Type: Calculate
  • Duration: 30min
  • Credits: 5
  • Difficulty: 0.6
  • Tags: hpi improper integral
  • Note:
    HPI, 2014-05-26, Mathe 2, Aufgabe 32
  • Created By: ad-si
  • Created At:
    2014-07-26 15:31:39 UTC
  • Last Modified:
    2014-07-26 15:31:39 UTC