Let I=∫_0^1 arctan (x)/x d x

3. Let

   $$ I=\int_0^1 \frac{\arctan (x)}{x} d x $$

   (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

   $$ \left|\int_0^1 \frac{\arctan (x)}{x} d x-\int_0^1 \frac{P_2(x)}{x} d x\right| \leq \frac{1}{9} $$

   (c) Hence calculate $I$ with an error up to $\frac{1}{9}$.

高等数学 · 已解决 · 大学数学
提问于2023年05月08日 · 阅读 161