Zadanie 2f, Paper 3, listopad 2025, TZ3, Matematyka AA IB HL

Let $a=\sqrt2-1=\tan(\pi/8)$ and $T_N(x)=\sum_{k=0}^{N-1}(-1)^kx^{2k+1}/(2k+1)$. Define $I=\int_0^a2x\arctan x\,dx$. You may use this theorem: for an alternating series whose term magnitudes decrease to zero, the absolute error after finitely many terms is at most the magnitude of the next term.