A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

133. ∫ (sin²x) / (1 + sin²x) dx

Evaluate the improper integrals in Exercises 53–62.

∫ from 3 to ∞ of (2 / (u² − 2u)) du

Evaluate the improper integrals in Exercises 53–62.

∫ from 0 to 3 of (1 / √(9 − x²)) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

127. ∫ (ln x) / (x + x ln x) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x dx / √(2 − x)

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(x + 1) / (x² (x − 1))] dx