In indefinite integrals, the foundation of all foundations and most of the computation is the substitution method. Generally, it’s about converting the indefinite integral of one variable, e.g., 
, into the indefinite integral of another variable, e.g., 
, with the latter ideally being easier to compute, of course.
The eternal question is: what should I substitute for ? What substitution should I use?
The answer is not thrilling…
If you are just starting your journey with indefinite integrals, I have bad news for you. The news is this: there’s no template.
The rule is that there is no rule. There are no keys like: “substitute for what’s under the square root or what’s in the denominator” or “if it’s this then I do this, if it’s different then I do something else.”
Accept this.
Only after solving about 20 integrals will you develop some intuition and sense, and then it will be time to start figuring out “what I need to take so that the derivative of this includes the function under the integral sign” and so on.
Guess, guess, guess…
What should you do if you haven’t yet solved those 20 indefinite integrals? It’s simple. You need to guess and check if the substitution fits. Definitely on paper!!!! Not in your head or from memory!!! If the substitution doesn’t fit? Tough, cross it out and try another.
There’s nothing worse than staring at an indefinite integral, straining your brain – without even picking up a pen – then resignedly shaking your head and saying, “I don’t know what to substitute here”.
Instead, write immediately and guess various substitutions that come to your mind. No one is out to harm you, so you shouldn’t have to wait long to find the correct substitution, that is, one that replaces all the variables with variables .
You can find many examples of how to start in my Indefinite Integrals Course.
Good luck!
