Question

Solve $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\left(2{\sec }^{2}x+x^3+2\right)dx$

Answer 1

$I={\int }_0^{\left(\frac{\pi }{4}\right)}(2se{c}^{2}x+x^3+2)dx$

$={(2tanx+(\frac{x^4}{4})+2x)}_0^{\left(\frac{\pi }{4}\right)}$

$=\left[2tan\right(\frac{\pi }{4})+(\frac{1}{4}\left)\right(\frac{\pi }{4}{)}^4+2\times \left(\frac{\pi }{4}\right)]-[0+0+0]=2+\left(\frac{{\pi }^4}{1024}\right)+\left(\frac{\pi }{2}\right)$