A Fascinating Integral

This is a simplified version of an article that appeared in The Mathematical Gazette July 1995*

 


ó
õ

1

 

0

x4

(

1-x

)

4

 

 

1+x2

dx=

22

7

-p

 


The simplest way to prove this is to divide the polynomials to get:

ó
õ

1

 

0

x4

(

1-x

)

4

 

 

1+x2

dx

=

ó
õ

1

 

0

æ
ç
ç
è

x6-4x5+5x4-4x2+4-

4

1+x2

ö
÷
÷
ø

dx

 

=

é
ê
ê
ë

1

7

x7-

2

3

x6+x5-

4

3

x3+4x-4arctan x

ù
ú
ú
û

1

 

0

 

=

æ
ç
ç
è

1

7

-

2

3

+1-

4

3

+4-4

p

4

ö
÷
÷
ø

-

(

0

)

 

=

22

7

-p

 
As well as this astonishing result one can also use it to find the value of p correct to 2 decimal places:

since x4( 1-x) 4/1+x2>0 for 0<x<1, it follows that p <22/7.

Furthermore, for 0£ x£ 1

x4

(

1-x

)

4

 

 

1+x2

£

x4

(

1-x

)

4

 

 

 

=

[

x

(

1-x

)

]

4

 

 

 

£

é
ê
ê
ë

1

2

æ
ç
ç
è

1-

1

2

ö
÷
÷
ø

ù
ú
ú
û

4

 

 

 

=

1

28

 

=

1

256

 
since the maximum value of x( 1-x) occurs when x=1/2 (why?)

Hence

ó
õ

1

 

0

x4

(

1-x

)

4

 

 

1+x2

dx

£

ó
õ

1

 

0

1

256

dx

 

=

1

256

 

<

0.004

 
and so

22

7

-0.004<p <

22

7

which means that 22/7 is an approximation to p accurate to two decimal places.

All this from one integral!

The graph of f(x)=x4( 1-x) 4/1+x2 is itself of interest, being very flat in the interval [ 0,1] . Hence the function is called a pancake function

f(x)=x4( 1-x) 4/1+x2Graphic: graph.gif

*Pancake functions and approximations to p , The Mathematical Gazette, Vol 79, Number 485, July 1995 pp371-374, which won an award for the best Note in 1995.


This document was translated from LATEX by HEVEA.

Scientific Notebook was used to write LATEX.