Suppose \(0 \le f \le g\) on \([a,b)\) and that \(\lim_{x \to
b^-}f(x) = +\infty\text{.}\) Show that for improper integrals \(\int_a^b
f(x)dx\) and \(\int_a^b g(x)dx\) the following relations hold (c.f. Proposition 4.11)
if \(\int_a^b g(x)dx\) converges, so does \(\int_a^b f(x) dx\text{.}\) Moreover, \(\int_a^b f(x)dx \le \int_a^b
g(x)dx\text{.}\)
if \(\int_a^b f(x)dx\) diverges to \(+\infty\text{,}\) then so does \(\int_a^b g(x)dx\text{.}\)
Formulate and proof a similar statement about the improper integrals for functions \(0 \le f \le g\) on \((a,b]\) with \(\lim_{x
\to a^+} f(x) = +\infty\text{.}\)
2.
Show that for all \(n \ge 0\text{,}\)\(\int_0^{\infty} x^n e^{-x}dx = n!\) (Hint: choose \(f(x) = x^n\) in Exercise 3.5.6.
3.
Determine the values of \(p\) for which the integral \(\int_0^1
x^p\ln(x) dx\) converges. And for those values of \(p\text{,}\) find the integral. (Hint: use integration by parts.)
4.
Use the comparison test to determine the convergence of the integral
Factor \(x^b\) out from the square root. Then make a suitable substitution to convert the integral into one that can be handled by trigonometric substitution.
