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### How to show analytically that $x^4-4x^3-2x^2+16x+24=0$ has no solutions in $\mathbb{R}$?

$$f(x)=x^4-4x^3-2x^2+16x+24=(x^2-2x-4)^2+2x^2+8>0.\tag{1}$$ My motivation: the terms $-4x^3$ and $16x$ are annoying since they can be either positive or negative, so I wanted to put them in a ...
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### Can we compute this integral $\int_0^{2\pi} \frac{1}{5+3 \cos x} dx$ using an antiderivative defined on $\mathbb R$?

Your first antiderivative is "correct", that isn't (quite) the problem. When I say "correct" rather than correct, the point is that to speak accurately we should always talk about ...
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