<s>[INST] Maximum $\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$ where $y\in \left[0,1\right]$? How find maximum this integral $$\int_{0}^{y}\sqrt{x^4+\left(y-y^2\right)^2}dx$$ where $y\in \left[0,1\right]$?
 [/INST] HINT:
By taking derivative with respect to $y$ and setting it to be equal to $0$ we have :
$$\int_{0}^{y}\frac{(y-y^2)(1-2y)}{\sqrt{x^4+(y-y^2)^2}}dx+\sqrt{y^4+(y-y^2)^2}=0$$
Hence:
$$\int_{0}^{y}\frac{dx}{\sqrt{x^4+(y-y^2)^2}}dx=\frac{\sqrt{y^4+(y-y^2)^2}}{(y-y^2)(2y-1)}$$
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