Integral and Derivative

Let $C = \displaystyle\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(x) dx$ , which is a constant. Then

$f'(x) = \tan^{2}(x) + C = \sec^{2}(x) - 1 + C \Longrightarrow f(x) = \tan(x) - x + Cx + D$

for some constant $D$ . We are then given that

$f\left(\dfrac{\pi}{4}\right) = \tan\left(\dfrac{\pi}{4}\right) - \dfrac{\pi}{4} + \dfrac{\pi}{4}C + D = -\dfrac{\pi}{4} \Longrightarrow \dfrac{\pi}{4}C + D = -1$ , (i).

Now when we integrate $f(x) = \tan(x) - x + Cx + D$ from $-\dfrac{\pi}{4}$ to $\dfrac{\pi}{4}$ , since $\tan(x)$ and $x$ are odd functions they will contribute nothing to this symmetric integral, so we are left to evaluate $Dx$ from the lower to upper bound, giving us that $C = \dfrac{\pi}{2}D$ . Plugging this result into equation (i) yields that

$\dfrac{\pi}{4} \times \dfrac{\pi}{2}D + D = -1 \Longrightarrow D\left(1 + \dfrac{\pi^{2}}{8}\right) = -1 \Longrightarrow D = -\dfrac{8}{8 + \pi^{2}} \Longrightarrow$

$C = \dfrac{\pi}{2}D = -\dfrac{4\pi}{8 + \pi^{2}} = -0.703226$ to 6 decimal places.

Thus $\lceil -1000 \times C \rceil = \lceil 703.226 \rceil = \boxed{704}$ .