In a summative assessment, I lost a mark due to this:
$f_X(x)=\frac{\lambda}{\sqrt{2\pi}}\int_{-\infty}^\infty y^2\exp\left\{-\left(\frac{1}{2}+\lambda x\right)y^2\right\} \; dy$
Now let $a=\frac{1}{2}+\lambda x$ for brevity and $u=ay^2$ so $du=2ay \; du$
Then
$f_X(x)=\frac{2\lambda}{\sqrt{2\pi}}\int_0^\infty \frac{u}{a} e^{-u} \; du.$
The comment was "show that you have considered any possible change in limits", and "y=" and "u=" were inserted into my script, so that it now reads:
$f_X(x)=\frac{\lambda}{\sqrt{2\pi}}\int_{y=-\infty}^{y=\infty} y^2\exp\left\{-\left(\frac{1}{2}+\lambda x\right)y^2\right\} \; dy$
Then
$f_X(x)=\frac{2\lambda}{\sqrt{2\pi}}\int_{u=0}^{u=\infty}\frac{u}{a} e^{-u} \; du.$
This is not a calculus module (it's statistical theory, final year undergraduate), and I feel pretty hard done by ! Is this overly pedantic on the part of the marker ?
EDIT, rather than correct the typos in this post, which would obscure robjohn's answer, I have posted an answer which (hopefully) doesn't contain any typos and makes the point more clearly.