Exercise
$\int_1^{e^{-x}}\left(\sqrt{1+v^3}\right)dv$
Step-by-step Solution
1
The integral $\int\sqrt{1+v^3}dv$ is non-elementary
$\left[\frac{2}{5\sqrt{v^3+1}}\left(v^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(v+1\right)}\sqrt{v^2-v+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(v+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+v\right)\right]_{1}^{e^{-x}}$
Final answer to the exercise
$\left[\frac{2}{5\sqrt{v^3+1}}\left(v^4+\sqrt[6]{-1}\sqrt[4]{\left(3\right)^{3}}\sqrt{\sqrt[6]{-1}\left(v+1\right)}\sqrt{v^2-v+1}F\left(\arcsin\left(\frac{\sqrt{-\sqrt[6]{\left(-1\right)^{5}}\left(v+1\right)}}{\sqrt[4]{3}}\right)\Big\vert \sqrt[3]{-1}\right)+v\right)\right]_{1}^{e^{-x}}$