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Define $A+B=\{a_{i}+b_{i}|1\le i\le n\}$,where $A=\{a_{1},a_{2},\cdots,a_{n}\},B=\{b_{1},b_{2},\cdots,b_{n}\}$,
Let $B$be a permutation of $A=\{-3,-2,-1,0,1,2,3\}$
I found there exsit $B$ such $A+B=A$.
For eaxmple:$$A=\{-3,-2,-1,0,1,2,3\},B=\{0,1,3,-2,2,-1,-3\}$$
since
$$-3,-2,-1,0...
What kind of real analysis tools would you employ for this integral?
$$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$
EDIT: Here is a supplementary question, the cubic log version
$$\int_0^1 \frac{z \log ^3\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$
Lemma:
If $f \in C([a,b])$ and $\int_a^b f(x) h(x) dx=0 \ \forall h \in C^2([a,b])$ with $h(a)=h(b)=0$ then $f(x)=0 \ \forall x \in [a,b]$.
Proof of lemma:
Suppose that there is a $x_0 \in (a,b)$ such that $f(x_0) \neq 0$, for example without loss of generality we suppose that $f(x_0)>0$.
Bec...
