a distance 
from the start. What is the maximum time it could have taken to travel the distance 
is called a Jane's integer if 
for some positive integers 
and 
.
there exist infinitely many positive integers 
such that the set of 
consecutive
contains exactly 
Jane's integers.
Y by Adventure10, Mango247
Without using nummerical calculators, figure out the order to maximize and minimize the integral below.
![\[ \int_{0}^{\pi/4} a ( \, b( \,c \, ) \, ) \,dx \]](//latex.artofproblemsolving.com/5/9/2/59247f36a35c8d40067ecc1e22c987d792227b85.png)
Where
and 
can be switched with 
and 
You are only allowed to use each trig function once. Two examples are given below
![\[ \int_{0}^{\pi/4} \cos ( \, \sin( \,\tan(x) \, ) \, ) \,dx \qquad \text{and} \qquad \int_{0}^{\pi/4} \sin ( \, \cos( \,\tan(x) \, ) \, ) \,dx\]](//latex.artofproblemsolving.com/d/d/1/dd1e61fb4d7d47bbfc8b15c60f77c2eccd9f02df.png)
You are to figure out the order of these trigonometric functions to find the minima and maxima of the integral. No calculators allowed.
![\[ \int_{0}^{\pi/4} a ( \, b( \,c \, ) \, ) \,dx \]](http://latex.artofproblemsolving.com/5/9/2/59247f36a35c8d40067ecc1e22c987d792227b85.png)
Where
and 
can be switched with 
and 
You are only allowed to use each trig function once. Two examples are given below
![\[ \int_{0}^{\pi/4} \cos ( \, \sin( \,\tan(x) \, ) \, ) \,dx \qquad \text{and} \qquad \int_{0}^{\pi/4} \sin ( \, \cos( \,\tan(x) \, ) \, ) \,dx\]](http://latex.artofproblemsolving.com/d/d/1/dd1e61fb4d7d47bbfc8b15c60f77c2eccd9f02df.png)
You are to figure out the order of these trigonometric functions to find the minima and maxima of the integral. No calculators allowed.
