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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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    -1: I don't see any mathematical question here. Besides, what does it mean to be acceptable? Without context, this is not a real question. Voting to close as NARQ.2011-02-07
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    Here's an analogy. Problem: Solve x^2+x-90=0. Response: Look in back of textbook, where is says that $x=9$ and $x=-10$ are the answers. Solution: $9^2+9-90=0$ and $(-10)^2-10-90=0$, so sure enough, $x=9$ and $x=-10$ are the answers. Would you consider this a valid method of equation solving? Checking your work is an important step, but there is a lot more to finding antiderivatives than that.2011-02-07
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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 have been voting to undelete and reopen all reasonable questions, but they need more votes. Please peruse the list of Undelete and Reopen items in the 10K tools and help reverse the destructive campaign. See Review/Tools/{Close,Delete} Thanks.2014-08-05
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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