My math is rusty, and although I initially thought I understand the solution, upon further examination I think I don't: That's the original function:
$$ \Psi(x,t) = A \mathrm{e}^{-\lambda|x|} \mathrm{e}^{-\mathrm{i} \omega t} $$
\begin{align*} \langle x^2 \rangle &= 2|A|^2 \int_0^\infty x^2 \mathrm{e}^{-2\lambda x}\,\mathrm{d}x \\ &= 2 \lambda \left[ \frac{2}{(2\lambda)^3} \right] \\ &= \frac{1}{2\lambda^2} \text{.} \end{align*}
What I don't understand is why there's 2 in front of A square, why parameters of integration changed from minus infinity-plus infinity to 0-plus infinity, and why x lost its absolute value. At first I thought that he's using the symmetry of the function and calculating the integral from 0 to infinity, where |x| = x, then multiplying it by two. But after checking how to integrate absolute value functions I'm not sure my reasoning is correct.
Sorry if the question is messed up, but I cannot imbed images directly, and have to use links.
Your reasoning seems correct to me. There's some dot/inner product of functions defined there (in your particular context).
https://mathworld.wolfram.com/InnerProduct.html
They just use this particular definition of the inner product, and calculate $\langle x,x\rangle$ i.e. $\langle x^2\rangle$
The rest of your reasoning is OK, indeed that's why all these modifications took place.