One of my student asked me to help her evaluate this indefinite integral $$\int\dfrac{\cos x}{1+e^x}\mathrm{d}x,$$ and I tried several minutes, but at last I had to given up, for I thought that it is very possible the primitive of $\dfrac{\cos x }{1+e^x}$ can not be expressed in terms of elementary functions. And then, I resorted to Maple and Mathematica. But these two computer algebra systems can not give me the answer, which shows that it is very certain that that indefinite integral is irreducible. But since I am not very familiar with differential Galois theory, I do hot know how to tell my student? Can anyone help me?
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Another way to ask this question, is to ask which functions cannot be integrated? – Don Larynx Dec 21 '14 at 05:11
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You may suggest that it is a difficult question, and this advanced theory is to find for example here: http://math.stackexchange.com/a/1075296/72361 – Przemysław Scherwentke Dec 21 '14 at 05:24
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1@Don : I do not understand the motivation behind making this question as a duplicate. Also, why are we marking duplicates to duplicates? I have travelled through one too many a Duplicaception tunnel. – Nick Dec 21 '14 at 15:38
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Since: $$\int e^{-kx}\cos x\,dx = \frac{\sin x - k\cos x}{(1+k^2)\,e^{kx}}$$ we just have: $$\int\frac{\cos x}{1+e^x}\,dx = \sum_{k\geq 1}(-1)^{k+1}\frac{\sin x - k\cos x}{(1+k^2)\,e^{kx}}.$$
Jack D'Aurizio
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@Nick: the first formula follows from integration by parts and the second one by expressing $\frac{1}{1+e^x}$ as $e^{-x}-e^{-2x}+e^{-3x}-\ldots$. – Jack D'Aurizio Dec 21 '14 at 12:19