What do we mean by convergent and divergent integrals?

Question

aravindg · Answer

@ganeshie8

ganeshie8 · Answer

convergent means the definite integral evaluates to a fixed number
divergent means it evaluates to infinity

anonymous · Answer

One converges and one diverges..? One has a finite limit, the other has an infinite limit..? One evaluates to a number and the other does not( it has an infinite limit approaching one of the bounds).

aravindg · Answer

How do we identify if an integral is convergent or divergent?

ganeshie8 · Answer

evaluate the integral and see 
for ex :-

$\large  \int_1^{\infty}   \frac{1}{x} dx$

aravindg · Answer

[log x] from 1 to infinity 

It is infinity.

ganeshie8 · Answer

yes, so we say integral is divergent

 it has no finite value

aravindg · Answer

I see. What is the relation of a convergent integral with limits?

ganeshie8 · Answer

it is useful in infinite series

ganeshie8 · Answer

integral test is one of the convergence tests.. 

if the integral representing the series converges/diverges, then the corresponding sum also converges/diverges

aravindg · Answer

Could you provide an example of using limits to solve a convergent integral?

ganeshie8 · Answer

oh you're asking about using limits in evaluating the integral.. 
one sec let me cookup some example quick.. :)

ganeshie8 · Answer

try this : 

$\large \int_1^{\infty} \frac{1}{x^2} dx $

ganeshie8 · Answer

and this  :

$\large  \int_0^1 \frac{1}{\sqrt{x}}dx$