Question Number 104609 by ย Mยฑth+et+s last updated on 22/Jul/20

prove:  ∫_0 ^1 ((x^4 (1−x)^4 )/(1+x^2 ))dx=((22)/7)−π

Answered by Dwaipayan Shikari last updated on 22/Jul/20

∫_0 ^1 ((x^4 (1−2x+x^2 )^2 )/(1+x^2 ))dx  ∫_0 ^1 ((x^4 ((1+x^2 )^2 −4x(1+x^2 )+4x^2 ))/(1+x^2 ))dx  ∫_0 ^1 x^4 (1+x^2 )−4x^5 +((4x^6 )/(1+x^2 ))dx  ∫_0 ^1 x^4 +x^6 −4x^5 +((4x^6 +4)/(1+x^2 ))−(4/(1+x^2 ))dx  [(x^5 /5)+(x^7 /7)−((4x^6 )/6)]_0 ^1 +∫4(x^4 −x^2 +1)−[4 tan^(−1) x]_0 ^1   (1/5)+(1/7)−(2/3)+(4/5)−(4/3)+4−4.(π/4)  =1+(1/7)−2+4−π  =3+(1/7)−π  =((22)/7)−π     It proves that 𝛑 is smaller than ((22)/7)

Commented byDwaipayan Shikari last updated on 22/Jul/20

๐Ÿ˜ƒ๐Ÿ˜›

Commented byAr Brandon last updated on 22/Jul/20

Great decomposition ๐Ÿ‘

Commented byย Mยฑth+et+s last updated on 22/Jul/20

nice work sir

Commented byDwaipayan Shikari last updated on 22/Jul/20

๐Ÿบ๐Ÿป๐Ÿ˜ƒ