Answer
$$256 - 62\sqrt {17} $$
Work Step by Step
$$\eqalign{
& \int_0^1 {\frac{{6{x^3}}}{{\sqrt {16 + {x^2}} }}} dx \cr
& {\text{Let }}x = 4\tan \theta ,{\text{ }}dx = 4{\sec ^2}\theta d\theta \cr
& \theta = {\tan ^{ - 1}}\frac{x}{4} \cr
& {\text{The new limits of integration are:}} \cr
& x = 1 \to \theta = {\tan ^{ - 1}}\frac{1}{4} \cr
& x = 0 \to \theta = {\tan ^{ - 1}}\frac{0}{4} = 0 \cr
& {\text{Substituting}} \cr
& \int_0^1 {\frac{{6{x^3}}}{{\sqrt {16 + {x^2}} }}} dx = \int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {\frac{{6{{\left( {4\tan \theta } \right)}^3}}}{{\sqrt {16 + {{\left( {4\tan \theta } \right)}^2}} }}} \left( {4{{\sec }^2}\theta } \right)d\theta \cr
& = \int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {\frac{{6{{\left( {4\tan \theta } \right)}^3}}}{{\sqrt {16 + 16{{\tan }^2}\theta } }}} \left( {4{{\sec }^2}\theta } \right)d\theta \cr
& = \int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {\frac{{6{{\left( {4\tan \theta } \right)}^3}}}{{4\sqrt {{{\sec }^2}\theta } }}} \left( {4{{\sec }^2}\theta } \right)d\theta \cr
& = \int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {\frac{{6{{\left( {4\tan \theta } \right)}^3}}}{{\sec \theta }}} \left( {{{\sec }^2}\theta } \right)d\theta \cr
& = 384\int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {{{\tan }^3}\theta } \left( {\sec \theta } \right)d\theta \cr
& {\text{Recall that }}{a^{m + n}} = {a^m}{a^n} \cr
& = 384\int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {{{\tan }^2}\theta } \left( {\sec \theta \tan \theta } \right)d\theta \cr
& = 384\int_0^{{{\tan }^{ - 1}}\frac{1}{4}} {\left( {{{\sec }^2}\theta - 1} \right)} \left( {\sec \theta \tan \theta } \right)d\theta \cr
& {\text{Integrating}} \cr
& {\text{ = 384}}\left[ {\frac{1}{3}{{\sec }^3}\theta - \sec \theta } \right]_0^{{{\tan }^{ - 1}}\frac{1}{4}} \cr
& {\text{ = 384}}\left[ {\frac{1}{3}{{\sec }^3}\left( {{{\tan }^{ - 1}}\frac{1}{4}} \right) - \sec \left( {{{\tan }^{ - 1}}\frac{1}{4}} \right)} \right]{\text{ - 384}}\left[ {\frac{1}{3}{{\sec }^3}\left( 0 \right) - \sec \left( 0 \right)} \right] \cr
& {\text{Where }}\sec \left( {{{\tan }^{ - 1}}\frac{1}{4}} \right) = \frac{{\sqrt {17} }}{4} \cr
& {\text{ = 384}}\left[ {\frac{1}{3}{{\left( {\frac{{\sqrt {17} }}{4}} \right)}^3} - \frac{{\sqrt {17} }}{4}} \right] - {\text{384}}\left[ {\frac{1}{3} - 1} \right] \cr
& {\text{Simplifying}} \cr
& {\text{ = 384}}\left[ {\frac{{17\sqrt {17} }}{{192}} - \frac{{\sqrt {17} }}{4}} \right] + 256 \cr
& {\text{ = }}34\sqrt {17} - 96\sqrt {17} + 256 \cr
& {\text{ = }}256 - 62\sqrt {17} \cr} $$
