$\underset{\ln \lambda }{\overset{\ln \left(\frac{1}{\lambda }\right)}{\int }}=\frac{f\left(\frac{x^2}{3}\right)(f\left(x\right)+f(-x))}{g\left(3x^2\right)(g\left(x\right)-g(-x))}dx=$

$$\begin{gathered} \underset{\ln \lambda }{\overset{\ln \left(\frac{1}{\lambda }\right)}{\int }}=\frac{f\left(\frac{x^2}{3}\right)(f\left(x\right)+f(-x))}{g\left(3x^2\right)(g\left(x\right)-g(-x))}dx \\ \underset{\ln \lambda }{\overset{-\ln \lambda }{\int }}\frac{f\left(\frac{x^2}{3}\right)(f\left(x\right)+f(-x))}{g\left(3x^2\right)(g\left(x\right)-g(-x))}dx \\ f\left(\frac{x^2}{3}\right), \left(f\left(x\right)+f(-x)\right), g\left(3x^2\right)are \\ even function but g\left(x\right)-g(-x) odd function \\ hence \frac{f\left({\displaystyle \frac{x^2}{3}}\right)\left(f\left(x\right)+f(-x)\right)}{g\left(3x^2\right)\left(g\right(x)-(g(-x))}is odd function \\ I=0 \end{gathered}$$