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can i say that, if $M>0$ ^and $n\in \mathbb{N}$ $\displaystyle\left (\int_{-M}^{M} x^n \right)^{1/n} \leq\displaystyle\int_{-M}^{M} x$?

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  • $\begingroup$ is it $x^n dx $. $\endgroup$ – hamam_Abdallah Dec 3 '17 at 21:37
  • $\begingroup$ the right hand side is zero if you are integrating in $x$ $\endgroup$ – qbert Dec 3 '17 at 21:40
  • $\begingroup$ For $n$ even your LHS is positive while RHS is zero. For $n$ odd, both sides are zero. $\endgroup$ – Macavity Dec 4 '17 at 1:52
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Rather, we have $\left(M^{-1}\displaystyle\int_{0}^{M}x^{1/n}dx\right)^{n}\leq M^{-1}\displaystyle\int_{0}^{M}xdx$, this is known as Jensen's inequality applied to the convex function $x\rightarrow x^{n}$, $n\geq 1$.

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