Question

Evaluate the definite integral ${\int }_0^{\frac{\pi }{4}}(2{\sec }^{2}x+x^3+2)dx$

Answer 1

Let $I={\int }_0^{\frac{\pi }{4}}(2{\sec }^{2}x+x^3+2)dx$
$\Rightarrow \int (2{\sec }^{2}x+x^3+2)dx=2\tan x+\frac{x^4}{4}+2x=F\left(x\right)$
By second fundamental theorem of calculus, we obtain
$I=F\left(\frac{\pi }{4}\right)-F\left(0\right)$
$=\{(2\tan \frac{\pi }{4}+\frac{1}{4}{\left(\frac{\pi }{4}\right)}^4+2\left(\frac{\pi }{4}\right))-(2\tan 0+0+0\left)\right\}$
$=2\tan \frac{\pi }{4}+\frac{{\pi }^4}{4^5}+\frac{\pi }{2}$
$=2+\frac{\pi }{2}+\frac{{\pi }^4}{1024}$