Answer
$${\text{False}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\sqrt 3 } {\frac{{dx}}{{{{\left( {1 + {x^2}} \right)}^{3/2}}}}} \cr
& {\text{Let }}x = \tan \theta ,{\text{ }}dx = {\sec ^2}\theta d\theta \cr
& {\text{The new limits of integration are}} \cr
& x = 0,{\text{ }}\theta = {\tan ^{ - 1}}0 = 0 \cr
& x = \sqrt 3 ,{\text{ }}\theta = {\tan ^{ - 1}}\sqrt 3 = \frac{1}{3}\pi \cr
& {\text{Substituting}} \cr
& \int_0^{\sqrt 3 } {\frac{{dx}}{{{{\left( {1 + {x^2}} \right)}^{3/2}}}}} = \int_0^{\frac{1}{3}\pi } {\frac{{{{\sec }^2}\theta d\theta }}{{{{\left( {1 + {{\tan }^2}\theta } \right)}^{3/2}}}}} \cr
& = \int_0^{\frac{1}{3}\pi } {\frac{{{{\sec }^2}\theta d\theta }}{{{{\left( {{{\sec }^2}\theta } \right)}^{3/2}}}}} \cr
& = \int_0^{\frac{1}{3}\pi } {\frac{1}{{{{\left( {\sec \theta } \right)}^{1/2}}}}} d\theta \cr
& = \int_0^{\frac{1}{3}\pi } {\sqrt {\cos \theta } } d\theta \cr
& {\text{The statement is False}} \cr} $$
