This book is mainly written for someone who has completed a first calculus course, and assumes that you know how to symbolically integrate expressions of one variable. If you are taking your first calculus course right now, however, you can still use this book as a companion. I recommend you only use the techniques in this book to do math that you have already mastered in your coursework. This is to help make sure you learn the math well: when working some math problem that is new to you, writing out all the steps is the best way to cement your understanding and ingrain the correct habits. Once you have that clarity, you have a great foundation on which to do that kind of abstract math mentally. If you have any questions about this process, please show this book to your instructor and ask them for guidance.
Several places in this book refer to another book, called Inner Algebra, which teaches how to do algebra mentally. If you do not happen to have it, you can read the full text online at http://hilomath.com/inneralgebra/html/ .
While the aim of this book is to make you skillful at doing abstract math in your head, please know that it is not an all-or-nothing affair. At any time when doing mental math, you can always choose to write down a step to make sure you are clear. The most important thing when doing math is to maintain your understanding - never forget that!
If you have questions or comments, I'm happy to hear from you. Contact me by sending email to amax@hilomath.com.