For u-substitution, what is the mathematical systematic way of picking a u?

What is the systematic way of figuring out what u should be set equal to?

In class we were introduced to u-substitution, which is the chain rule in reverse. You pick a u that is the inner function find the derivative, substitute to make solving integrals easier. My only problem is in the complex integrals he solved, it seemed as if he just randomly picked what he wanted u to equal and it magically worked. Is there a step I am missing? Maybe, it is really obvious to figure out what u should be, but I don't know the really obvious way to tell. What is the systematic way of figuring out what u should be set equal to?

• If you see that there is a function $\phi$ such that $f(x)\ dx = f(\phi(t))\phi'(t)\ dt$ then of course pick $u=\phi(t)$. Otherwise, you just try to pick a $u$ such that the integral will look slightly simpler and then proceed from there (possibly using more substitutions down the line).
– user137731
Dec 13, 2015 at 16:13
• @Bye_World Ok, I see. It is usually hard to find the $\phi \left(t\right)$ though. But I guess I will get better at it. Sometimes if I pick a u that looks slightly simpler, I still end up at a dead end. Dec 13, 2015 at 16:17
• That happens to all of us. Hopefully after you've used this technique enough times you'll get better at deciding what a useful $u$-sub will be.
– user137731
Dec 13, 2015 at 16:19

$u$-substitution is basically the chain rule in reverse.

Just like using the chain rule, you want to pick judiciously, and you'll learn how to do this by doing lots of problems and seeing some common patterns come up.

• I was hoping for a more systematic approach. Yeah, I am suppose to try to find the inner function by looking at the integral, but sometimes that is a hard task to do. Sometimes it is more obvious, but most of the time it is actually really hard to find. I guess, you are right when saying I should just do a lot of problems. Thanks for the answer though. I hopefully will be able to see some common patterns that will come up. Dec 13, 2015 at 16:21
• Yep. Integration tends to be an art -- a good portion of the battle is seeing the problem in the right way. Dec 13, 2015 at 16:24

The systematic approaches are nearly impossible to describe - it takes books and books and books, relying on mathematics well above standard calculus, especially when you consider that there are innumerable integrals that don't have solutions in terms of standard elementary functions.

There are several processes in mathematics where forward is easy but reverse is hard. That, in fact, is what allows encryption on the web to work. When you get into differential equations, often times the main point is to setup the problem, not solve it, or to use some method to come up with an estimated solution rather than an exact one.

The best way to be able to pick a "u" is to have your derivative table memorized, and be well-practiced with complex derivatives and what they look like. This helps you "see" what you should pick for a "u". If you apply the chain rule a lot, it is then much easier to see what it might look like in reverse.

But, as Batman said, it tends to be as much as an art as anything.

If you want to know more about how a computer might do it, you should pick up a book on computer algebra algorithms. However, there are integrals that are solvable that are beyond the reach of humans, too. As I said, forward is simple but reverse is hard.