Improper Integral

Improper Integral

1. problem location is behaviour at infinity

   • first kind
     • positive integrand
       • increasing in x
         • bounded
           • convergent
             • limit exists
               • limit = sup {F(x)}
         • not bounded
           • divergent
             • no limit
     • integrands with variable signs
   • Third kind
     • consider separately and independently

2. problem location is behaviour at the point t=b

   • Second kind
     • removable discontinuity

3. Test Functions

4. Convergence test

   • Direct Comparison test
     • DCT could be inconclusive, careful to choose the right test function
   • Limit Comparison Test
     • 2-way limit comparison test
       • for improper integrals of the first kind
         • consider the dominant term
       • for improper integral of the second kind
         • problem location is t=b
         • problem location is t=a
     • 1-way limit comparison test
       • 0-finite rule
       • finite -finite rule
   • absolute convergence
   • Dirichlet Test

5. derivative jointly continuous

   • change of order

6. Leibnitz Rule

7. dominated convergence

   • absolute convergence
   • integral of the max of the absolute function convergence
   • rip-off method
   • envelope method

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