example of random variable that is integrable but have infinite second moment

Suppose $\Pr(X>x) = \dfrac 1 {x^\alpha}$ for $x\ge 1$.

Then $f_X(x) = \alpha x^{-\alpha-1}$ for $x\ge 1$.

So $\operatorname{E}(X) = \dfrac \alpha {1-\alpha}$ if $\alpha>1$.

And $\operatorname{E}(X^2) = \dfrac \alpha {2-\alpha}$ if $\alpha>2$.

If $1<\alpha<2$ then $\operatorname{E}(X)<\infty$ and $\operatorname{E}(X^2) = \infty.$

Now let's try a similar one with integer values in the set $\{1,2,3,\ldots\}$.

$$ \text{Let } \zeta(\alpha) = \sum_{n=1}^\infty n^{-\alpha}. $$

$$ \Pr(X=n) = \dfrac{n^{-\alpha}}{\zeta(\alpha)}.$$

(This is sometimes called a Zipf distribution, after George Kingsley Zipf.)

As above, $\operatorname{E}(X)<\infty$ if $\alpha>2$ and $\operatorname{E}(X^2) = \infty$ if $\alpha>3$.