Question

How do you find the definite integral of `1/x^2` from 0 to 1?

Answer 1

As this function is not continuous in the in the interval [0,1], it is not integrable in the interval [0,1].

Answer 2

This is an improper integral because the function is not defined at one of the limits of integration.

To (attempt to) evaluate the integral use the replacement:

`int_0^1 1/x^2 dx = lim_(ararr0^+) int_a^1 1/x^2 dx`

(The "`=`" is "equals if the limit exists")

`int_0^1 1/x^2 dx = lim_(ararr0^+) int_a^1 1/x^2 dx`

`color(white)"ssssssssss"` `= lim_(ararr0^+) (-1)/x ]_a^1`

`color(white)"ssssssssss"` `= lim_(ararr0^+) (-1+1/a) = oo`

The integral diverges. (Does not exist.)