Calculus II, Spring 2026 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework 01

1. Section 8.8:

Verify the convergence/divergence of the improper integrals ∫_{0}^{1} \frac{1}{x^{p}} dx and ∫_{1}^{∞} \frac{1}{x^{p}} dx for p > 1, p = 1 and 0 < p < 1, respectively by direct evaluation (total 6 cases). Then memorize the results.

2. Section 8.8:

Problems: 7, 13, 25, 31.

3. Section 8.8: Problems: 35, 39, 41, 45, 55, 65, 66.

4. Section 8.8:

Let f(x) = \frac{1}{x^{0.5} ln x}. Is the improper integral ∫_{e}^{∞} f(x) dx convergent? Hint: Try not to find the anti-derivative of f(x), but instead find a suitable g(x) for comparison using Theorem 2 or Theorem 3. See also Example 5-7 of Lecture 02 on how to find g(x).

5. Section 8.8:

Do the same for ∫0^{\frac{1}{e}} \frac{1}{x^{2}| ln x|} dx.