Evaluate the definite integral: $\int_{0}^{\frac{\pi}{4}}\left(2 \sec ^{2} x+x^{3}+2\right) d x$

Let $I = \int_{0}^{\frac{\pi}{4}} (2 \sec^2 x + x^3 + 2) dx$.
First,find the indefinite integral:
$\int (2 \sec^2 x + x^3 + 2) dx = 2 \tan x + \frac{x^4}{4} + 2x = F(x)$.
By the second fundamental theorem of calculus,$I = F\left(\frac{\pi}{4}\right) - F(0)$.
$F\left(\frac{\pi}{4}\right) = 2 \tan\left(\frac{\pi}{4}\right) + \frac{1}{4}\left(\frac{\pi}{4}\right)^4 + 2\left(\frac{\pi}{4}\right) = 2(1) + \frac{\pi^4}{4 \times 256} + \frac{\pi}{2} = 2 + \frac{\pi^4}{1024} + \frac{\pi}{2}$.
$F(0) = 2 \tan(0) + \frac{0^4}{4} + 2(0) = 0 + 0 + 0 = 0$.
Therefore,$I = 2 + \frac{\pi}{2} + \frac{\pi^4}{1024}$.