Exercise 5

Consider the definite integral $\int_0^1 x \tan(x) \, dx$.

1. Explain why this integral cannot be evaluated exactly by using either $u$-substitution or by integrating by parts.

2. Using appropriate subintervals, compute $L_4$, $R_4$, $M_4$, $T_4$, and $S_8$.

3. Which of the approximations in (b) is an over-estimate to the true value of $\int_0^1 x \tan(x) \, dx$? Which is an under-estimate? How do you know?