Aug 1, 2020 Thread starter Admin #1 anemone MHB POTW Director Staff member Feb 14, 2012 3,642 Find all continuous functions $f:[1,\,8] \rightarrow \mathbb{R} $ such that $\displaystyle \int_1^2 f^2(t^3)dt + 2\int_1^2 f(t^3)dt=\dfrac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt$

Find all continuous functions $f:[1,\,8] \rightarrow \mathbb{R} $ such that $\displaystyle \int_1^2 f^2(t^3)dt + 2\int_1^2 f(t^3)dt=\dfrac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt$