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I have a problem for a long time now which is I always make small stupid errors and mistakes in calculations when the question is easy or. This issue caused me a lot of problems in exams that I was known for having bad grades when the exam is easy and a good grades when the exam is challenging (I make a lot of mistakes when the question is easy for some reason).

Some examples:

When I first learned the Beta function $\displaystyle B(a,b):=\int_0^1 t^{a-1}(1-t)^{b-1}dt \ \ \ $ st $a,b >0$ I saw this identity $\displaystyle B(a,b)= \int_0^\infty \frac{t^{a-1}}{(1+t)^{a+b}}dt$ and I tried to prove it: Let $t= \frac{1}{u}-1 \ , dt =\frac{-du}{u^2}$. and $u=\frac{1}{1+t}$ $$ \int_0^\infty \frac{t^{a-1}}{(1+t)^{a+b}}dt =-\int_1^0 \frac{\left(\frac{1}{u}-1 \right)^{a-1}}{\left(\frac{1}{u} \right)^{a+b-2} }du =\int_0^1 {\color{red}{(u-1)}^{a-1} u^{b-1} } du $$ here I spent more than an hour trying to find the mistake

In his integral $\lim\limits_{n \to \infty} \int_0^ \infty \frac{nx\arctan{x}}{(1+x)(x^2+n^2)}dx$ I made a mistake $\int_0^ \infty {\arctan(x)}dx = \frac{\pi}{2}$ I couldn't find the error so I asked a question here and I was surprised how I even made that mistake. I was able to give a solution for this limit here.

In this integral $\displaystyle \int_{-\infty}^ \infty \frac{dx}{x^2+ 2x \cos(t)+1 } dx$ I made a mistake thinking that $\sec(\cos(x))=x$ and it took me about an hour to correct this error.

There are a lot of these examples that I made a stupid mistake in the easy part of the question, this happen to me very often That I once got $11/20 $ on a test, that test every one got bad grades because there was some challenging problems , but because of some stupid mistake I got the hard one correctly and pretty much all easy ones were wrong. My teacher was surprised how could one solve the hard problems and get most of the easy one wrong. There are a lot of other examples like this and this happen to me a lot and all my efforts or methods I used didn't help very much. I got very upset with this lately so I decided to ask here if someone has a solution to this problem.

pie
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  • Slowing down (both when you are writing and in your thinking) might help. Giving yourself more time allows you to carefully consider each step you are taking. – K. Jiang Mar 22 '24 at 07:39

1 Answers1

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It probably doesn't help to think in terms of hard problems versus easy problems. That probably means problems that require some conceptual thought versus relatively straightforward calculations. But being good at calculation is a skill in itself, and as your experience shows, it can't be taken for granted. Indeed, a distinguishing feature of some great mathematicians is how good they are at calculation.

Once you frame things that way, that there is skill that needs to be improved and can be learned, that's half the battle. It might have a lot to do just with making your writing better organized. Or when you think you have must have made a mistake when evaluating an integral, for example, can you run a plausibility check on each step? For example, for the question about $\arctan$, a quick check of what the graph of $\arctan$ looks like would have told you that your memory of its integral was wrong. Then you would have surely thought about it a bit more, and realized you were confusing it with $\tan$.

David M
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