Question

Find the value of the integral that converges.
∫^-5_-[infinity] x^-2 dx.

Answer 1

`\int_(-\infty)^(-5) x^(-2)dx= (1)/(5) + \lim_(x\to -\infty) (1)/(x) =(1)/(5)`

Because the
`\lim_(x\to -\infty) (1)/(x) =0`

The integral converges to
`(1)/(5)`

Explanation:

For this case we want to find the following integral:

`\int_(-\infty)^(-5) x^(-2)dx`

And we can solve the integral on this way:

`\int_(-\infty)^(-5) x^(-2)dx= (x^(-2+1))/(-2+1) \Big|_(-\infty)^(-5)`

`\int_(-\infty)^(-5) x^(-2)dx= -(1)/(x) \Big|_(-\infty)^(-5)`

And if we evaluate the integral using the fundamental theorem of calculus we got:

`\int_(-\infty)^(-5) x^(-2)dx= (1)/(5) + \lim_(x\to -\infty) (1)/(x) =(1)/(5)`

Because the
`\lim_(x\to -\infty) (1)/(x) =0`

The integral converges to
`(1)/(5)`