Evaluate the definite integral _0^ 4 ( 2 ^2 x + x^3 + 2 )dx - Vidyadip

Question

Evaluate the definite integral $\int\limits_0^{\frac{\pi }{4}} {\left( {2{{\sec }^2}x + {x^3} + 2} \right)dx} $

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Answer

$\int\limits_0^{\frac{\pi }{4}} {\left( {2{{\sec }^2}x + {x^3} + 2} \right)dx} $
$= 2\int\limits_0^{\frac{\pi }{4}} {{{\sec }^2}xdx + \int\limits_0^{\frac{\pi }{4}} {{x^3}dx + 2\int\limits_0^{\frac{\pi }{4}} {1dx} } } $
$= 2\left( {\tan x} \right)_0^{\frac{\pi }{4}} + \left( {\frac{{{x^4}}}{4}} \right)_0^{\frac{\pi }{4}} + 2\left( x \right)_0^{\frac{\pi }{4}}$
$= 2\left( {\tan \frac{\pi }{4} - \tan {0^o}} \right) + \frac{{{{\left( {\frac{\pi }{4}} \right)}^4}}}{4} - 0 $ $+ 2\left( {\frac{\pi }{4} - 0} \right)$
$= 2\left( {1 - 0} \right) + \frac{{\left( {\frac{{{\pi ^4}}}{{256}}} \right)}}{4} + \frac{{2\pi }}{4}$
$= 2 + \frac{{{\pi ^4}}}{{1024}} + \frac{\pi }{2}$
$= \frac{{{\pi ^4}}}{{1024}} + \frac{\pi }{2} + 2$

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