Use the properties of integrals to verify the inequality

Question

∫ sin(x)/√x dx ≤√2/2 from [π/4 , π/2]

Answer 1

The maximum of sin(x) / sqrt(x) can be found by setting its derivative equal to zero.

The derivative is cos(x) / x^.5 - (1/2) sin(x) / x^1.5 _.Setting this to zero results in the transcendental equation

2x = tan(x). The approximate solution is x = 1.1656

sin(x)/x^.5 evaluated at this point is approximately 0.8510. This is an absolute maximum on the interval

[π/4 , π/2 ]

Therefore the integral (over π/4 to π/4) of sin(x) /x^.5 ≤ .8510 * ( π/2 - π/4) = .6683

Finally .6683 is less than sqrt(2) /2 = .7071, so the inequality is proved.

This analysis has the shortcoming that the transcendental equation cannot be solved exactly. However, an approximate solution to any desired accuracy can be obtained by the Newton / Raphson method. So, in principle, the inequality can be proved with any precision desired.

There is probably a more elegant approach, but I was not able to find it quickly.