Going from to is adding NOT .
because and for all m.
The limits for the second transformation are wrong aren't they ?
Your problem, as I see it, is caused by your use of the same symbol for both the original and the transformed variable. What you have done, is to use the transformation for the integrand, but the transformation for the limits.
You seem to be going out of your way to confuse yourself, and you're still making the same mistake.
For your last calculation, you are now using as the 'original' variable and as the 'transformed' variable and the relationship between them is
For the stuff to the right of the integral sign is being replaced by
If you use the same replacement method for the limits, as you must, then you have to say that,
when then is given by That is,