I've seen cases while doing integrals and even on a question here where, if you're computing some integral I, you end up with $$I=I$$ Apart from the linked question, where this situation isn't really addressed as a problem in itself, I arrived there while trying to do, for example, $$\int x\csc 2x \mathrm dx$$ Is this a failure of the method itself or is there something wrong I'm doing?
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It means neither. It just means that's not a way you can reach the answer. Integration by parts doesn't always work, and that is not a failing of the method. And you getting $I = I$ means you most likely did things correctly.
You could try integrating by different parts. Or you could try a different approach. Or maybe the function just can't be antidifferentiated.
Arthur
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In some cases, the antiderivative of a certain function cannot be expressed as a combination of elementary functions like \begin{equation*} \sin x,\ln x,\arccos x,e^{x} \end{equation*} This does not necessarily mean that the method of integration by parts has failed, it's just said that these functions don't possess an elementary antiderivative.
Like for example, the function you gave does have an indefinite integral - you can find it here - https://www.wolframalpha.com/input/?i=integrate+x%2Fsin%282x%29. Check if you can express it in terms of elementary functions!
Ishraaq Parvez
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It's important to note, as Arthur did, that merely failing to express an integral in terms of elementary functions does not mean it's impossible, by parts or otherwise. – Patrick Stevens Apr 04 '21 at 12:09