Exercises

Exercises 4.3 Exercises

1.

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 4.12)

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

\begin{equation*} \int_0^{\pi/2} \frac{1}{x\sin(x)} dx \end{equation*}

🔗

5.

For \(0 \le b \lt 2 \text{,}\) show that

\begin{equation*} \int_0^1 \frac{dx}{\sqrt{x^b -x^2}} = \frac{\pi}{2-b} \end{equation*}

🔗

Hint.

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.

🔗

Prev Top Next