Find the value of `int_(0)^(2pi)1/(1+tan^(4)x)dx`

Answer

`pi`

Explanation

`I=int_(0)^(2pi)1/(1+tan^(4)x)dx=2int_(0)^(pi)(dx)/(1+tan^(4)x)`
`=4int_(0)^(pi//2) (dx)/(1+tan^(4)x)`……………….1
`=4int_(0)^(pi//2)(dx)/(1+cot^(4)x)`…………….2
Adding 1 and 2 we get
`2I=4int_(0)^(pi//2)dx=4,(pi)/2`
`impliesI,pi`