use a trigonometric substitution to find integral of 1/(1+3x^2)^3/2 between limits 0 and 1.
I think you have to use tan or sin not quite sure how though.
You can leave $\displaystyle x=\frac{1}{\sqrt{3}}tan({\theta}), \;\ dx=\frac{1}{\sqrt{3}}sec^{2}{\theta}d{\theta}$
This changes your limits of integration to 0 to Pi/3.
Making the subs will whittle it down to:
$\displaystyle \frac{1}{\sqrt{3}}\int_{0}^{\frac{\pi}{3}}{cos{\th eta}}d{\theta}$