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In school we have just started with integration by parts. We had examples like $∫x\sin(x)\,dx$ or $∫x^2\sin(x)\,dx$ I asked myself if it is possible to integrate terms like $∫x^{25}\sin(x)\,dx$ without doing integration by parts 25 times. I can't tell how exactly I did it, but I integrated some explicit terms and created this: $$\int x^n\sin x\,dx=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^{k+1}x^{n-2k}{n!\over(n-2k)!}\cos x+\sum_{k=0}^{\lfloor(n-1)/2\rfloor}(-1)^kx^{n-2k-1}{n!\over(n-2k-1)!}\sin x$$

with $n\in \Bbb N$.

I tested it a few times and I think that it is correct, but I don't have any idea how to prove it. How can I do that?

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    If an integration formula is correct, you can prove it by differentiating.2012-11-05
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    I took the liberty to type in your formula in LaTeX. Check if I haven't made some mistakes. Also, if anyone has a better idea how to type it in in terms of formatting, feel free to do so (is there a good way to type in formulae as long as this?).2012-11-05
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    Alternatively, are you familiar with proof by mathematical induction? You could try to prove it by induction on $n$.2012-11-05
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    Nice job, Gunnar. I think you are a future mathematician.2012-11-05
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    Those interested in this question should see also http://math.stackexchange.com/questions/231100/indefinite-integral-of-xn-cdot-sinx-proof2012-11-06

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