[Calculus] Curious Integral of Johann Bernoulli
By: Tao Steven Zheng (郑涛)
【Problem】
In 1697, the Swiss mathematician Johann Bernoulli (1667 - 1748) discovered a very interesting result of integral calculus:
Prove this interesting result!
Hints
(1) Begin by using the transformation .
(2) Remember the Taylor series expansion .

【Solution】
This integral is an improper integral with the indeterminate form . It is well known that
and knowing this fact enables us to show that
Applying the Taylor series gives
Now use the technique of integration by parts on the integral . Let
,
,
, and
.
Subsequently,
Continuing the integration by parts scheme to the integral gives the pattern:
Thus,
Consequently,
Therefore,
