Bounds Integral of x times arcsine

Homework Statement

Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16

The Attempt at a Solution

Not sure what to bound with. Do we use Squeeze Theorem?

andrewkirk
Homework Helper
Gold Member
The curves of ##y=x## and ##y=sin^{-1} x## start with the second below the first, then intersect so the second is above the first, then meet again at ##x=1##.

You can get a lower (upper) bound by integrating the square of the lower (higher) of the two from 1/2 up to the first intersection point, then doing the same again from that point up to 1 (noting that the lower and upper will have switched at the first intersection).

Wolfram tells me that ##(sin^{-1} x)^2## has a closed form antiderivative, so you should be able to obtain all derivatives.

That will give you upper and lower bounds, but I don't know if they are narrow enough to meet the problem spec. Worth a try anyway.

EDIT: Ignore this. Vela's approach is much easier. The above gives much tighter bounds but that is not required by the question. I was a little concerned about the degree of difficulty in this solution.

Last edited:
vela
Staff Emeritus
Homework Helper

Homework Statement

Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16

The Attempt at a Solution

Not sure what to bound with. Do we use Squeeze Theorem?
Over the interval given, what are the upper and lower bounds of ##\arcsin x##?

andrewkirk