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%  Homework for the course "Calculus II", 
%  Spring semester, 2026, Wei-Cheng Wang.

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\noindent
{\scriptsize Calculus II, Spring 2026 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

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\large
Homework 01   
\normalsize
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%           The questions!

% {\bf Part 1}:
\begin{enumerate}

% Section 8.8: % Improper integrals
\item \label{xp}
Section 8.8: % Improper integrals

Verify the convergence/divergence of the improper integrals 
$\displaystyle{\int_0^1 {1\over x^{p}} dx}$
and
$\displaystyle{\int_1^\infty {1\over x^{p}} dx}$ 
for $p>1$, $p=1$ and $0<p<1$,
respectively by direct evaluation (total 6 cases).
Then memorize the results.

% Section 8.8: % Improper integrals
\item
Section 8.8: 

Problems: 7, 13, 25, 31. %35, 39, 41, 45, 55, 65, 66, 77, 81, 83.
% Problems: 7, 13, 19, 21, 25, 31, 35, 39, 42, 45, 46, 55, 65, 66, 77, 81, 83.
% problem 77 is harder,need more hint

% Section 8.8: % Improper integrals
\item
Section 8.8: 
Problems: 35, 39, 41, 45, 55, 65, 66.%, 77, 81, 83.
% Problems: 7, 13, 19, 21, 25, 31, 35, 39, 42, 45, 46, 55, 65, 66, 77, 81, 83.
% problem 77 is harder,need more hint
% Hint for problem 77:

% Section 8.8: % Improper integrals
\item
Section 8.8: 

Let $f(x) = \ds{1 \over x^{0.5} \ln x}$.
Is the improper integral $\ds{\int_{e}^\infty f(x) \ dx}$ 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 $g(x)$.

% Section 8.8: % Improper integrals
\item
Section 8.8: 

Do the same for $\ds{\int_0^{1\over e} \ds{1 \over x^2 |\ln x|} \ dx}$.

\end{enumerate}

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