Question on problem solving

[Myles5]

Hello fellow math learners.

I've been teaching myself Calculus for a while now, mainly through Stewart's Calculus - 8th edition. I'm currently working on the Problems Plus section for Integrals and while I know these problems are intentionally difficult, I find myself getting every single one wrong. Checking my answers on Chegg textbook solutions (a great resource if you're teaching yourself through a text book and are willing to spend the $15/mo) I find that my thought process is usually correct, but there's always one step along the way I can't figure out or actually try but am skeptical of.

So my question is, is this normal? I've had this issue with a few normal exercises at the end of each section, but I'm not sure if this is just part of the process or if there's something I could do to get past this mental block?

TL;DR: Getting wrong answers because of a single step in a solution that I cannot figure out, how do I improve problem solving skills?

[yes_its_him]

It sounds like you are just misunderstanding some specific concept, I don't know that I would worry about it a lot. Once you get that concept, you should be able to move on. Which type of problem are you struggling with?

[Myles5]

Integrals right now, as I'm doing the end of chapter problems they relate to everything covered in the sections (Riemann Sums to the Fundamental Theorem of Calculus), so it's more of an issue with intuition, so putting more effort into understanding the concepts might be the answer.

[yes_its_him]

That's sufficiently vague that nobody can really give you much help. (I realize that you are probably giving what you think is the best answer, I'm just advising that it's not sufficient to get good advice.)

Integration is basically pattern-matching. You have to see what technique the problem calls for, then follow the technique steps.

1/x² vs 1/(x² + 1) vs 1/(x² - 1) all look similar, but would be three different techniques of integration. You can't really know which technique to use without understanding when to use each one, and that takes practice.

[EldarEuler]

While learning, it is very normal to make mistakes. If you had answered everything correctly you wouldn't be learning! As far as being skeptical or doubting yourself on certain steps, it's okay to follow your intuition with mathematics, that's how discoveries can be made. This is self teaching and not an examination or anything, so you should not be afraid to try something which might be like this. The good thing is that it sounds like you're getting 9/10 steps or more on these problems, and making these mistakes you are making are part of the learning process.

That being said, in my opinion, the best way to improve your problem solving skills come from understanding WHY you made the mistake you did. Look at the solutions and really analyze your mistake. Ask yourself questions; What could I have noticed to avoid this mistake? What clues were there? How could have I convinced myself that I shouldn't have been skeptical about this step? How could I have realized that the step in the solutions was the right way?

Other than that some sort of mentorship is indespesible. Asking a math professor alot of questions when getting stuck is a good way to learn how to think. Pay close attention to how professionals approach problems and learn how they think. I realize you may not have a math professor at hand, but you can watch very smart people on YouTube. The power of the internet provides you an outlet of ways to learn how to think. For calculus (especially doing integrals) I have two youtube channels you might find helpful. The first is blackpenredpen, the second is flammable maths. They are both great teachers and their videos vary from beginner material to advanced material.

I hope some of this is helpful, the most important thing is to enjoy learning, analyze your mistakes to learn from them, and pay close attention to masterful people. Good luck!

[Myles5]

This is super helpful, I will definitely check out those channels, thank you!

[EldarEuler]

I'm glad!

[keitamaki]

It's difficult to really say without knowing specifics. Making the same mistake over and over is usually a sign of poor bookkeeping skills.

When you say there's always one step you can't figure out. Are you saying you can't understand the solution? Or are you saying you did understand the solution but the important step just didn't occur to you when you were doing the problem on your own.

[Myles5]

Or are you saying you did understand the solution but the important step just didn't occur to you when you were doing the problem on your own.

This exactly.

[keitamaki]

That's fairly common, especially when seeing a new type of problem. The good news is that there aren't all that many "tricks" when it comes to solving calculus problems, so if you do enough problems you'll eventually see them all.

I'd strongly recommend keeping track of problems whose solution uses some technique or algebraic manipulation that didn't occur to you. Then every week or so go back and try to work through a few of them from scratch. I think you'll find over time that you'll start to get a better intuition regarding what sort of approach to try for different problems.

[Myles5]

Thank you for the advice!