Reddit mentions of Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (2nd Edition)

Your question title focuses on indefinite integrals, but from what you wrote it sounds like what's really confusing you is the use of differentials. I think to really understand stuff about differentials, you're going to want to read about the theory of differential forms. I haven't really read this stuff yet myself, but here are a few places for you to look:

• Hubbard/Hubbard - Vector Calculus, Linear Algebra, and Differential Forms
  
• Spivak - Calculus on Manifolds
  
• Bachman - A Geoemetric Approach to Differential Forms
  
• Chapter 10 of Rudin's Principles of Mathematical Analysis
  
  The short answer is that differentials are used very haphazardly in elementary calculus, sometimes called notation, and other times used for intuition.. and occasionally put on a semi-rigorous foundation. But I think to really get what's going on, and see it developed in an actually rigorous way, you need to learn the concepts from one of the books above.
  
  Of course, before you can really attempt most of the content above, you'd need to understand multivariable calculus and analysis.. which could take a while. But since Rudin is such a classical text and he develops the material from the beginning, that book might be your best bet.

Barbara Burke Hubbard, John H. Hubbard:
Vector Calculus, Linear Algebra and Differential Forms A Unified Approach (1998)

Contents: http://i.imgur.com/1Hj4h52.png

Amazon

Other people may recommend other versions but the important part is the author Hubbard.

Some possibilities:

Calc I & II: Spivak's Calculus

Calc III and a bit of linear algebra: Hubbard & Hubbard's Vector Calculus

LA: Axler or Shilov or both

ODE: Morris Tanenbaum


Discrete/Combinatorics/etc.: Knuth's Concrete Mathematics

For book suggestions beyond concerning Analysis, Algebra, and Topology, the search box will turn up a ton of previous conversations.

Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, by Hubbard and Hubbard.

I would like to say as a physicist that this book contains everything I have ever needed to know save contour integration. Calculus, linear algebra, and differential forms are extremely well tied together, and extremely important concepts such as Lagrange multipliers are developed as applications of the core material. Amazing book.