If ${\displaystyle \int f\left(x\right)dx=f\left(x\right)}$ , then ${\displaystyle \int {\left\{f\left(x\right)\right\}}^{2}dx=}$

Now ${\displaystyle \int {\left\{f\left(x\right)\right\}}^{2}dx={\displaystyle \int f\left(x\right).f\left(x\right)dx}}$

$={\displaystyle \int f\left(x\right).f^1\left(x\right)dx}$

$f\left(x\right)=t$

$f^1\left(x\right)dx=dt$

$={\displaystyle \int tdt}$

$=\frac{t^2}{2}=\frac{{\left\{f\left(x\right)\right\}}^2}{2}$

$=\frac{1}{2}{\left\{f\left(x\right)\right\}}^2$