I realized after graphing sin(x) and sin(x^2) that the first curve is still "on the way up" to its first peak, but the second one has already passed it's first peak and is coming down. If you draw a horizontal line through the intersection point, then that horizontal line will intersect y=sin(x) somewhere to the right.
By symmetry, those 2 points are equidistant from x=pi/2, which is the x value where y=sin(x) peaks.
And the two x values of the two intersections of the horizontal line are x and x^2.
So the average value of x and x^2 is pi/2.
So: x^2 + x - pi = 0
so x =
[-1 + sqrt(1 + 4 pi)]/2 = 1.341627718...
[-1 - sqrt(1 + 4 pi)]/2 = -2.341627718...
Answer to our question: [-1 + sqrt(1 + 4 pi)]/2
edit: Jer beat me to it, but we got there slightly different ways
Edited on November 16, 2018, 2:02 pm
Posted by Larry
on 2018-11-16 13:59:14