Question

Prove the relationship:-

$\large 0 < \displaystyle\int_{ 0 }^{ 1 } \frac{ x ^ { 4 } (1-x) ^ { 4 } }{ 1+x ^ { 2 } } d x = \frac{ 22 }{ 7 } - \pi$


(Hint: This denotes $\large \pi = 3.14.....$ )​

Answer 1

Answer:

How is pi = 22/7 or 3.14?

Answer :

Twenty years ago, It was discovered that pi (π = 3.14…) is encoded in the Hellenic language. It is true that we have strong indications for the mathematical structure of the Hellenic language and also for its intelligence, since we do know that by using the ancient Hellenic alphanumeric system, we can receive the value of this mathematical constant with its first three decimals by the strict geometrical definition of it

The following relation gives the lexarithmic extraction of the mathematical constant π correct to three decimals:

(Length of circumference of circle)/(diameter) = = (338+1016+940)/730 = 2294/730 = 3,1424657534…!!!

The above value of π given by the lexarithmic extraction is a litlle more accurate than the Archimedean approximation 22/7 = 3.142857… and also the number of the fractional expression 2294/730 contains the numbers 22 and 7 in their correct positions in the numerator and the denominator…!

Answer 2

$\int _0^1\frac{x^4\left(1-x^4\right)}{1+x^2}dx=\frac{2}{35}\quad \left(\mathrm{Decimal:\quad }\:0.05714\dots \right)$

$\mathrm{Long\:division}\:\frac{x^4\left(1-x^4\right)}{1+x^2}:\quad -x^6+x^4$

$=\int _0^1-x^6+x^4dx$

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=-\int _0^1x^6dx+\int _0^1x^4dx$

$\int _0^1x^6dx=\frac{1}{7}$

$\int _0^1x^4dx=\frac{1}{5}$

$=-\frac{1}{7}+\frac{1}{5}$

$\mathrm{Simplify\:}-\frac{1}{7}+\frac{1}{5}:\quad \frac{2}{35}$

$=\frac{2}{35}$