Question

How do you test the improper integral `int x^-2dx` from `[-1,1]` and evaluate if possible?

Answer 1

The improper integral:

`int_(-1)^1 x^(-2)dx`

is divergent.

Explanation:

Consider first that `x^-2` is an even function:

`(-x)^-2 = 1/(-x)^2 = 1/x^2= x^(-2)`

so that:

`int_(-1)^1 x^(-2)dx = 2int_0^1 x^-2dx`

Now pose:

`f(t) = int_t^1 x^-2dx = [x^-1/(-1)]_t^1 = -1+1/t= (1-t)/t`

So that:

`int_0^1 x^(-2)dx = lim_(t->0^+) f(t) = lim_(t->0^+) (1-t)/t = +oo`

then the improper integral:

`int_(-1)^1 x^(-2)dx`

is divergent.