Question

Integrate (1+t 2 )dt/(1+t 4 ) Thank You.

Answer 1

let I be the given integral
dividing numerator and denominator by t^2 we get,
I = integral {1 + 1/t^2}/{t^2 + 1/t^2}dt
I = integral {1 + 1/t^2}/{t^2 + 1/t^2 - 2 + 2}dt
I = integral {1 + 1/t^2}/{(t-1/t)^2 +2}dt
let t-1/t = x
therefore (1 - 1/t^2)dt = dx
therefore I = integral dx/(x^2 +2)
I = arctan(x/(2)^1/2) * (1/2)^(1/2)
I = arctan((t-1/t)/(2)^1/2) * (1/2)^(1/2)