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6

Let $x=u^2$, $dx = 2 u \,du$. Then the integral is $$2 \int_0^1 \frac{du}{\sqrt{1-u^2}}$$ Now let $u=\sin{t}$, $du = \cos{t} \, dt$. Then the integral is $$2 \int_0^{\pi/2} dt \frac{\cos{t}}{\cos{t}} = 2 \frac{\pi}{2} = \pi$$

6

If to get from my home to the university I need to take a bus to Jerusalem, and then an internal bus; on the way back I first need to take an internal bus, and then a bus from Jerusalem. If you think about $AB$ as applying $B$ and then applying $A$, reversing it means first undoing $A$ and then undoing $B$. (Of course, as noted on this page, you need to ...

4

There is a combinatorial proof that is quite easy to follow (and remember, too). Assume that $G$ is a finite group and $p\mid |G|$. Let $X$ be the following set: $$X = \{ (g_1,g_2,\ldots,g_p) : g_1 g_2\cdot\ldots\cdot g_p = e\}.$$ We have $|X| = |G|^{p-1}$, since for every element of $X$ we are free to take the first $(p-1)$ coordinates as we want, then ...

3


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