The total number for distinct `x epsilon[0,1]` for which `int_(0)^(x)(t^(2))/(1+t^(4))dt=2x-1` is __________.

Options

A. 1
B. 2
C. 3
D. infinitely many

Answer

A

Explanation

Let `f(x)=underset(0)overset(x)int (t^(2))/(1+t^(4))dt=2x-1`,Clearly, f(x) is
`f'(x)=(x^(2))/(1+x^(4))-2=(1)/((1)/(x^(2))+x^(2))-2=(1)/((x-(1)/(x))^(2)+2)-2lt0`
So,f(x) is a strictly decreasing function on [0,1].
Also, `f(0)=1 gt0`
and, `f(1)=underset(0)overset(1)int (t^(2))/(1+t^(4))dt-1 lt 0[:'underset(0)overset(1)int (t^(2))/(1+t^(4))dt le underset(0)overset(1)int (1)/(2)dt=(1)/(2)]`
So,f(x)=0 has exactly one solution in [0,1].