How do I know which substitution to use if I am integrating by substitution?

It is not just trial and error, as this will eat away time in the exam. if you can't see it straight away then you need to consider some things. for example a common question might be find the integral of (3x² + 1)/(x³+x+9) dx. as you can imagine, this is quite tricky to find a substitution for. in fact here you don't even need one, there is a trick that integrating f'(x)/f(x) dx = ln|f(x)| + c. this here gives ln|x³+x+9| + c. A common rule for substituton for sin and cos also exists, whether to use double angle formula or to use sin²x + cos²x = 1. if the highest indeci is positive, you want to use double agnle forumla. if it is negative, use the other identity.  for t = tan(1/2 x) you are given the substitution in the question. for a hyperbolic function things get harder, the more you do the easier it will come. the main idea is that you have to be making things easier. if you remember that cosh²x - sinh²x = 1, then a²cosh²x - a²sinh²x = a². so if you have a function involving an a²+x² you should consider using x=asinhu (u is just a different variable we have introduced) as this will simplify to a²cosh²u. practice different integrals as this is the best way to remember which substitution to use.