Question Number 114699 by bemath last updated on 20/Sep/20

  ∫_0 ^(π/2)  (dx/( (√(1+tan ^4 x)))) ?

Commented byDwaipayan Shikari last updated on 20/Sep/20

Answered by bobhans last updated on 20/Sep/20

replacing x = (π/2)−x  I=∫_(π/2) ^0  ((−dx)/( (√(1+tan ^4 ((π/2)−x)))))  I=∫_0 ^(π/2)  (dx/( (√(1+cot ^4 x)))) = ∫_0 ^(π/2)  ((tan ^2 x)/( (√(tan ^4 x+1))))dx  2I=∫_0 ^(π/2)  ((sec ^2 x)/( (√(tan ^4 x+1)))) dx   I=(1/2)∫_1 ^∞  (dt/( (√(t^4 +1)))) ; [ t = tan x ]  set q = 1+t^4  ; t = (q−1)^(1/4)   I=(1/2)∫_1 ^∞  (1/( (√q))).(1/4)(q−1)^(−(3/4))  dq  I=(1/8)∫_1 ^∞  q^(−(1/2)) (q−1)^(−(3/4))  dq  I= (1/( 8))∫_1 ^∞ q^(−(5/4)) (1−q^(−1) )^(−(3/4)) dq  I=(1/8).((Γ^2 ((1/4)))/(Γ((1/2)))) = (1/(8(√π))).Γ^2 ((1/4))