Question Number 101601 by Dwaipayan Shikari last updated on 03/Jul/20

∫_((√2)−1) ^((√2)+1) ((x^4 +x^2 +1)/((x^2 +1)^2 ))dx

Answered by bemath last updated on 03/Jul/20

∫ (((x^2 +1)^2 −x^2 )/((x^2 +1)^2 )) dx = x−∫ (x^2 /((x^2 +1)^2 )) dx  I_2 = ∫ (x^2 /((x^2 +1)^2 )) dx   [ x = tan p ]   I_2  = ∫ ((tan ^2 p . sec ^2 p dp)/(sec ^4 p))  = ∫ tan ^2 p cos ^2 p dp   = ∫ ((1/2)−(1/2)cos 2p) dp  = (1/2)p −(1/4)sin 2p =(1/2)tan^(−1) (x)−(x/(2(x^2 +1)))  I= 2−(1/2)(tan^(−1) ((√2)+1)−tan^(−1) ((√2)−1))  −(1/2)((((√2)+1)/(4+2(√2))) −(((√2)−1)/(4−2(√2))))