When you can do the integral without it, I think! If I look at an integral and cannot see how it can be done without making one or another substitution, I proceed to make one.How to know when you DON'T need to use substitution (integrals)
We've been doing substitution for a few weeks now, and it's so embedded in my mind that when I look to take an integral, I immediately look for a substitution. I was just trying to do a simple problem and was looking at it for several minutes until I finally realized a substitution was completely unnecessary.
This may seem like a silly question, but what kinds of things separate having to use a u-substitution to not having to use it? What should I be looking for as quick indicators?
I'm not good at these things myself, but I think toying with different substitutions eliminates this problem.
Say, if you have , write it as and sub ; or
write it as and let ; or write it as to see
that no sub is needed. This way you'll learn when a sub is necessary and/or is more efficient than another.
There are no absolute rules for when a substitution will work. There are some tricky integrals out there where a clever substitution that seemingly comes out of thin air can be used to evaluate the integral.
That said, if you understand that performing a substitution is doing the chain rule backwards, then usually a substitution is used when you have a composition of 2 functions, and the derivative of the inner function is also inside the integrand. If you don't have this situation, then you probably want to try other methods before trying to find a clever substitution.
One more note: If you were to form an integral at random, then it is extremely unlikely that substitution would work.
One thing that you may find helpful is to write down a bunch of simple functions like etc. compose them, and then take the derivative of the inner function. You'll get a clear picture of when substitutions work.
Here's one example:
.
We get . By design this is a function which can be integrated by substitution.
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