Answer
$\frac{\pi}{6}$
Work Step by Step
$\int_{0}^{\frac{\sqrt 3}{2}}\frac{1}{1+4x^{2}}dx$
The pattern of the denominator $1+4x^{2}$ matches the pattern for $arctan$. The formula to solve to $arctan$ is:
$\int\frac{du}{a^{2}+u^{2}}=\frac{1}{a}arctan(\frac{u}{a})+C$
$u=2x$
$du=2dx$
$a=1$
$\frac{2}{2}\int_{0}^{\frac{\sqrt 3}{2}}\frac{1}{1+4x^{2}}dx$
$\frac{1}{2}\int_{0}^{\frac{\sqrt 3}{2}}2\frac{1}{1+4x^{2}}dx$
$\frac{1}{2}[\frac{1}{1}arctan(\frac{2x}{1})]_{0}^{\frac{\sqrt 3}{2}}$
$\frac{1}{2}[arctan(2(\frac{\sqrt 3}{2})-arctan(2)(0)$
$\frac{1}{2}[arctan(\sqrt 3)-arctan0]$
$\frac{1}{2}[\frac{\pi}{3}-0]$
$\frac{\pi}{6}$
