difficult integral

## Primary tabs

# difficult integral

Submitted by slashfan on Thu, 12/08/2005 - 21:23

Forums:

hello,

is is possible to show that integral of sinx.x^(-1/4) on [0,pi] is

less than pi^(3/4) and is it possible to find f that is square integrable such that (integral of (f-sinx)^2) <= 1 and

(integral of (f-cosx)^2) <= 4/9 on [0,pi]

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## Re: difficult integral

Part 1:

To begin, you might want to do the integral numerically and see how thevalue compares with the value you would like it to be. If so, you could look for some simple-to-integrate function like a piecewise linear function which is larger than your function but whose integral is less than your bound.

Part 2:

Think of this in terms of function space. sin and cos are two points in function space. Find the distance betwen these two points. You are looking to see if it is possible to find a point which is at most a certain distance from one point and some other distance from another point. This will possible if andonly if the triangle inequality is satisfied.

## Re: difficult integral

Part I:

integral of (sinx.x^(-1/4)) on [0,pi]=integral of (sinx.x^(-1/4)) on [0,pi/2] + integral of (sinx.x^(-1/4)) on [pi/2,pi].

but:

integral of (sinx.x^(-1/4)) on [0,pi/2] <= integral of (x.x^(-1/4)) on [0,pi/2] = (4/7)(pi/2)^(7/4),

and:

integral of (sinx.x^(-1/4)) on [pi/2,pi] <= integral of (sinx.(pi/2)^(-1/4)) on [pi/2,pi] = (pi/2)^(-1/4).

so:

integral of (sinx.x^(-1/4)) on [0,pi] <= (4/7)(pi/2)^(7/4) + (pi/2)^(-1/4) < (pi)^(3/4).

Part II:

nothing to add to what rspuzio mentioned. just note that because on [0,pi]: ||sinx-cosx||2 = (pi)^(1/2) > 1 + 2/3, the triangle inequality does not hold and so you can't find such a function.

good luck