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Integral calculus - Improper integral

  1. The Definite integral \(\displaystyle\int\limits_a^b f(x)dx\) is called an improper integral
    If
    1)a or b is infinite ,
    2) f(x) has one or more points of discontinuity
      in the interval [a,b]
  2.  if f(x) is a continuous function on [a, \(\infty\)), then 
    \(\displaystyle\int\limits_a^b f(x)dx =\lim_{n\to \infty} \int\limits_a^n f(x)dx\).

    Improper integral.
  3. if (f(x) is a continuous function on \(( -\infty, b],\) then 
    \(\displaystyle\int\limits_{-\infty}^b f(x)dx =\lim_{n\to -\infty} \int\limits_n^b f(x)dx\)

    Improper integral.
  4. \(\displaystyle\int\limits_{-\infty}^\infty f(x)dx =\int\limits_{-\infty}^ c f(x)dx+\int\limits_c^ \infty f(x)dx\)

    Improper integral.
    If for some real number c, both of the integrals in the right side are convergent, then the integral \(\displaystyle\int\limits_{-\infty}^\infty f(x)dx\) is also Convergent ; Otherwise it is divergent .
  5. Comparison Theorems
    Let f(x) and g(x) be continuous functions on the closed interval [a,\(\infty\)). Suppose that \(0 \leq g(x) \leq f(x) \) for all x in [a,\(\infty\)).
    1) \(\displaystyle\int\limits_a^\infty f(x)dx\) is convergent, then \(\displaystyle\int\limits_a^\infty g(x)dx\) is also convergent, 
    2)  \(\displaystyle\int\limits_a^\infty f(x)dx\) is divergent, then \(\displaystyle\int\limits_a^\infty g(x)dx\) is also divergent, 
  6. Absolute Convergent 
     \(\displaystyle\int\limits_a^\infty f(x)dx\) is convergent, then integraal  \(\displaystyle\int\limits_a^\infty g(x)dx\) is absolutely convergent
  7. Discontinuous Integrand 
    Let f(x) be a function which is continuous on the interval [a,b) but is discontinuous at x=b. then 
    \(\displaystyle\int\limits_a^b f(x)dx= \lim_{\epsilon \to0+} \int\limits_a^{b-\epsilon} f(x)dx\)

    Improper integral.
  8. Let f(x) be a continuous function for all real numbers x in the interval [a,b] except for some point c in (a,b). then
    \(\displaystyle\int\limits_a^b f(x)dx= \lim_{\epsilon \to0+} \int\limits_a^{c-\epsilon} f(x)dx + \lim_{\delta \to0+} \int\limits_{c+\delta}^b f(x)dx\)
    Improper integral.