$\int\limits_0^{\pi /4} {\left( {2{{\sec }^2}x + {x^3} + 2} \right)dx}$

: Let $I = \int\limits_0^{\pi /4} {\left( {2{{\sec }^2}x + {x^3} + 2} \right)dx} = \left[ {2\tan \,x + \cfrac{{{x^4}}}{4} + 2x} \right]_0^{\pi /4}$

$= 2\left( {\tan \cfrac{\pi }{4} - \tan 0} \right) + \cfrac{1}{4}\left( {\cfrac{{{\pi ^4}}}{{256}} - 0} \right) + 2\left( {\cfrac{\pi }{4} - 0} \right)$

$= 2\left( {1 - 0} \right) + \cfrac{{{\pi ^4}}}{{1024}} + \cfrac{\pi }{2} = \cfrac{{{\pi ^4}}}{{1024}} + \cfrac{\pi }{2} + 2$