3. Let
$I = \int_0^1 \frac{\arctan(x)}{x} dx$.
(a) Find the Taylor polynomial of order 2, $P_2(x)$, about $x = 0$ for the function $\arctan(x)$.
(b) Use Lagrange's formula for the remainder $R_2(x) = \arctan(x) - P_2(x)$ to show that
$| \int_0^1 \frac{\arctan(x)}{x} dx - \int_0^1 \frac{P_2(x)}{x} dx | \le \frac{1}{9}$
(c) Hence calculate $I$ with an error up to $\frac{1}{9}$.