# 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
$\displaystyle\int\limits_a^b f(x)dx =\lim_{n\to \infty} \int\limits_a^n f(x)dx$.

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

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

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
$\displaystyle\int\limits_a^b f(x)dx= \lim_{\epsilon \to0+} \int\limits_a^{b-\epsilon} f(x)dx$

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
$\displaystyle\int\limits_a^b f(x)dx= \lim_{\epsilon \to0+} \int\limits_a^{c-\epsilon} f(x)dx + \lim_{\delta \to0+} \int\limits_{c+\delta}^b f(x)dx$