I am writing a take home quiz for a first-year university level calculus course. This quiz deals with improper integrals. I would like to know if you would be kind enough to share your favorite improper integral. I am specially interested in :
- challenging thought provoking questions
- questions that can be tackled in various different methods, as found in a typical north america first year university level..., such as by-parts, u-subs, trig subs, trig ids, etc, etc..
I very much appreciate your input.
Not sure how much this qualifies for first-year, but perhaps clever first-year:
$$\int_0^{\infty} dx \frac{\log{x}}{x^2-1} = \frac{\pi^2}{4}$$
This one has a lot of nifty stuff built in. (Note that the integrand does not blow up at $x=1$.) What I would tell a first-year to do is sub $x=e^t$ and manipulate the integrand a bit to see that you can use a geometric series, which then becomes a well-known sum.
(Perhaps I would tell then to take as given that
$$\sum_{k=1}^{\infty} \frac1{n^2} = \frac{\pi^2}{6}$$ )
ADDENDUM
Another cool thing you can have a student show from this is that
$$\int_0^1 dx \frac{\log{x}}{x^2-1} = \frac{\pi^2}{8}$$
This may be shown by splitting the integral up at $x=1$ and applying the substitution $x=1/y$.
Once the student gets that, then it should be a breeze to prove that
$$\int_0^{\infty} dx \frac{\log{x}}{x^2+1} = 0$$