Integrate the function (1+x^3)^(1/2) from 0 to 2

 Exercise

\int_0^2\left(\sqrt[2]{1+x^3}\right)dx

 

The integral $\int\sqrt{1+x^3}dx$ is non-elementary

\left[\frac{2}{5\sqrt{x^3+1}}\left(x^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(x+1\right)}\sqrt{x^2-x+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(x+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+x\right)\right]_{0}^{2}

\left[\frac{2}{5\sqrt{x^3+1}}\left(x^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(x+1\right)}\sqrt{x^2-x+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(x+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+x\right)\right]_{0}^{2}

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