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

´int 1/sqrt(x) dx´

The integral of ´1/sqrt(x)´ is ´2 sqrt(x)´: ´= 2 sqrt(x) + "constant"´

´int_0^1 dx/sqrt(x) = (2 sqrt(1) + "constant") - (2 sqrt(0) + "constant") = 2´

Subtask 2

´int 1/(x-1)^2 dx´

For the integrand ´1/(x-1)^2´, substitute ´u = x-1´ and ´du = dx´: ´= int 1/u^2 du´

The integral of ´1/u^2´ is ´-1/u´: ´= -1/u + "constant"´

Substitute back for ´u = x-1´: ´= 1/(1-x) + "constant"´

´int_1^2 dx/(x - 1)^2 =´ The Integral does not converge!

Subtask 3

´int 1/x^2 dx = -1/x + "constant"´

´int_1^oo dx/x^2 = 1´


Solution
    1. ´2´
    2. The Integral does not converge!
    3. ´1´
  • URL:
  • Language:
  • 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: adius
  • Created At:
    2014-07-26 15:31:39 UTC
  • Last Modified:
    2014-07-26 15:31:39 UTC