Answer
$a.\quad\sinh^{-1}\sqrt{3}$
$b.\quad \ln(\sqrt{3}+2)$
Work Step by Step
$(a)$
Use table: "Integrals leading to inverse hyperbolic functions"
1. $\displaystyle \quad \int\frac{du}{\sqrt{a^{2}+u^{2}}}=\sinh^{-1}(\frac{u}{a})+C, \quad a\gt 0$
Here, $a=2$, $u(x)=x$.
$ \displaystyle \int_{0}^{2\sqrt{3}}\frac{dx}{\sqrt{2^{2}+x^{2}}}=\left[ \sinh^{-1}(\frac{x}{2}) \right]_{0}^{-2\sqrt{3}}$
$=\sinh^{-1}\sqrt{3}-\sinh^{-1}0\qquad $... ( $\sinh 0=0 )$
$=\sinh^{-1}\sqrt{3}$
$(b)$
Using the formulas in the box above these exercises,
$\sinh^{-1}x=\ln(x+\sqrt{x^{2}+1})$ ,$\quad -\infty \lt x \lt \infty$
$\sinh^{-1}\sqrt{3}=\ln(\sqrt{3}+\sqrt{3+1})=\ln(\sqrt{3}+2)$
