Question

Finding whether the following integral converges or diverges?

Answer 1

The integral converges

Explanation:

This is the graph of the function `y = frac(1)(sqrt(x^(5) + 2))`.

Clearly, from `x = 1` onwards, the area under the curve converges to a finite value.

Answer 2

Obviously

`int_1^oo 1/sqrt(x^5+2)dx >0`

For `x>1` we have

`x^4 < x^5+2 implies 1/x^2>1/sqrt(x^5+2)`

Thus

`int_1^oo 1/sqrt(x^5+2)dx < int_1^oo dx/x^2`

The latter integral is

`lim_{Lto oo} int_1^L dx/x^2 = lim_{L to oo} (1-1/L) = 1`

and thus we have

`0 < int_1^oo 1/sqrt(x^5+2)dx < 1`

and thus the integral converges.