Integration by parts and change of variables proofs with typos

[sponsoredwalk]

I'm just curious about the proofs of Integration by Parts & the Change of Variables formula
as given in this book on page 357. I think there are a lot of typo's so I've uploaded my
rewrite of them but I am unsure of how correct my rewrites are. If someone could point
out the errors & why I made them I'd really appreciate it as I don't see how I messed up
but still feel I did, for example I'm not sure whether the $$g_{i-1}$$ is a typo in
the last lines of the IBP proof, when the author takes the norm of the partition to zero.
I can't see how it's included according to my rewrite

 

[hunt_mat]

For integration by parts, why not just integrate the product rule for differentiation?

 

[sponsoredwalk]

The challenge I suppose. So no errors?

 

[hunt_mat]

I don't see why you just done use the fundamental theorem of calculus. That's what we did in our analysis course.