Proving Two Equations Involving nCr and Powers of 2

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Homework Statement


Prove:
The sum of r=0 to n [((-1)^r) * (nCr)] = 0.

Prove:
The sum of r=0 to n [nCr] = 2^n


Homework Equations


It says to consider (1-1)^n and (1+1)^n , but I have no idea what this is even relating to honestly.


The Attempt at a Solution


I need help getting started.
 
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You do know that
(a+ b)^n= \sum_{r=0}^n \left(\begin{array}{c}n \\ r\end{array}\right)a^{n-r}b^r
don't you?
 


Yes I do, but I don't know how to apply it. I realize the (1-1)^n is the considered piece for the alternating series. But do I just Induction with it?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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