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Let's say I got stuck with a problem in integration, and I decided to peek at an answer, and I differentiate the answer to get the question, and then I flip from question to answer, and I have my solution ready.

But, I have many times seen that this approach does not match with the way we find answer with question, it somehow looks illogical, because I feel like I had never seen the answer, then I would have never guessed this could be in my entire life.

Is this approach acceptable, or standard way? I would like to have your opinion!

Pardon me if I put the wrong type of question. And I couldn't find any tag for my question.

Regards,

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    "Valid" is context dependant. It is certainly a way in which correct answers are obtained; if *I* were grading, though, it would be an *invalid* way of obtaining the answer, since there is no work to justify testing $9$ and $-10$. The only person who can answer this question accurately is whoever is grading explorex's work. NARQ.2011-02-07

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Well, if you guess the correct answer, it's not wrong. In fact, it's important to have good guesses. But you can't guess a more complicated integral, so it's also important to know the integration methods.

Integration is an art. You got to have a bag of some known primitives, some tricks and some methods. What you have to do is combine them to get a solution.

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The idea is a reasonable one, and can be used even when there is no ``back of the book'' available. For example, suppose that we want to find $\int xe^x dx$. It is not unreasonable to guess that $xe^x$ might be an antiderivative. If we differentiate, we get $xe^x +e^x$: not quite right. But it is now easy to see that $xe^x -e^x$ will do the job.

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    I am not fully acquainted with the machinery. It seems senseless to haunt the "Review" list. We are mathematicians, and our goal here is to teach, and have a bit of fun besides. In a number of my answers, and very much more so in yours, a not so secondary goal is to teach the answerers. I can certainly vote to keep/reopen questions/answers that have value, and are under threat. The trouble is becoming aware of them, and assessing carefully. The destroyers operate under fewer constraints.2014-08-05