The total number of distinct `x in[0,1]` for which `int_0^x t^2/(1+t^4) dt =2x-1` is

Answer

1

Explanation

Let `f(x)=int_(0)^(1)(t^(2)dt)/(1+t^(4))-2x+1`
`implies f'(x)=(x^(2))/(1+x^(4))-2=(-2x^(4)+x^(2)-2)/(x^(4)+1)lt0,AAxepsilonR`
`:.f(x)` is decreasing function.
`f(0)=1` (from 1)
Now `0lt (t^(2))/(1+t^(4))lt1`
`:.0ltint_(0)^(1)(t^(2)dt)/(1+t^(4))lt1`
`:.f(1)=int_(0)^(1)(t^(2)dt)/(1+t^(4))-1lt0`
`:.` Graph of `y=f(x)` cuts x-axis exactly once for `x epsilon(0,1)`.
Hence only one solution.