Question

The total number of distinct $x\in (0,1]$ for which $\underset{0}{\overset{x}{\int }}\frac{t^2}{1+t^4}dt=2x-1\text{is}$

Answer 1

$\text{Let}f\left(x\right)=\underset{0}{\overset{x}{\int }}\frac{t^2}{1+t^4}dt-2x+1f^{\prime }\left(x\right)=\frac{x^2}{1+x^4}-2<0,\forall x\in [0,1]\Rightarrow f\left(x\right)\text{is decreasing function.}f\left(0\right)=1,f\left(1\right)=\underset{0}{\overset{1}{\int }}\frac{t^2}{1+t^4}dt-1\text{Now,}0\le \frac{t^2}{1+t^4}\le \frac{1}{2},\forall x\in [0,1]\Rightarrow \underset{0}{\overset{1}{\int }}\frac{t^2}{1+t^4}dt\le \underset{0}{\overset{1}{\int }}\frac{1}{2}dt\Rightarrow \underset{0}{\overset{1}{\int }}\frac{t^2}{1+t^4}dt-1\le \frac{1}{2}-1\Rightarrow f\left(1\right)<0$
$\therefore f\left(x\right)$ will cross x-axis exactly at one point for $x\in (0,1]$