Calculus Help

$1. \text{If}\ f(x) = x^2\ \text{and}\ g(x)=\ln(x), \text{compute } f'(1) + g'(1)$

$\text{We compute}\ f'(x) = 2x\ \text{and}\ g'(x)=\frac1x,\ \text{so plugging in 1 for both gives us}\ \boxed3.$

$2. \int_0^4\frac{dx}{\sqrt{|x-2|}}$

$\text{Since the function } \sqrt{|x-2|}\text{ discontinues at } x=2,\\ \text{we can split the integral into two parts and compute separately.}$

$\int_0^2\frac{dx}{\sqrt{|x-2|}}=\int_0^2\frac{dx}{\sqrt{2-x}}=-2\sqrt{2-x}|_0^2=2\sqrt2$

$\int_2^4\frac{dx}{\sqrt{|x-2|}}=\int_2^4\frac{dx}{\sqrt{x-2}}=2\sqrt{x-2}|_4^2=2\sqrt2$

$2\sqrt2+2\sqrt2=\boxed{4\sqrt2}$

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