Conditions

1. Infinite interval of integration.
   1. e.g. $\int_a^\infin f(x)dx$
2. discontinuity on the interior of the interval of integration
   1. e.g. $\int_a^b \frac {f(x)} c dx$ if a<c<b
3. Both 1 and 2 happens

Evaluation

Using limits to rewrite the integral as a proper integral.

• as for

  • If exist of every $\int_a^bf(x)dx$ exist of every b>a, then $\int_a^\infin f(x)dx=\lim_{b→\infin} \int_a^b f(x)dx$, provided the limit exists as is finite.
  • If exist of every $\int_b^cf(x)dx$ exist of every b<c, then $\int_{-\infin}^cf(x)dx=\lim_{b→-\infin}\int_b^c f(x)dx$, provided the limit exists as is finite.

  as for condition 2, split the integration to 2 parts that set the limit toward the x value of discontinuity.

  Convergence and Divergence

  • An Improper integral that equals a finite value is said to converge to that value.
  • An improper integral that does not equal a finite number is said to diverge.

Doubly Improper

Denoted as $\int_{-\infin}^\infin f(x) dx$, both sides are infinity.

like a bell curve, if both $\int_c^\infin f(x) dx$ and $\int_{-\infin}^c f(x) dx$ are converge, then $\int_{-\infin}^\infin f(x) dx$ is converge.

• sometimes, a doubly improper can be avoid by the considerations of symmetry. Just double it!

Multiple Evaluations Combined by Plus or Minus.

• just like this one: example problem

Imagine the integration as area under the curve

• if a finite area adds to a finite are will yield out a finite area, that is convergent
• if a infinite area adds to a finite are will yield out a infinite area, that is divergent
• if a infinite area adds to a infinite are will yield out a infinite area, that is divergent
• if a infinite area minus to a infinite are will yield out a infinite area, that is inderterminate.
  • for infinity have different levels and can’t simply cancel them out.

Tests and Concepts