Solve E = ∫ (from 0 to L) of (x^2+y^2)^-frac{3{2 wrt y

$$E = \int _ { 0 } ^ { L } ( x ^ { 2 } + y ^ { 2 } ) ^ { - \frac { 3 } { 2 } } d y$$

Solve for E

$$E=\frac{L}{\sqrt{x^{2}+L^{2}}x^{2}}$$

$$x\neq 0$$

Solve for L

$$\left\{\begin{matrix}L=-\frac{Ex^{3}}{\sqrt{1-E^{2}x^{4}}}\text{, }&\left(x>-\frac{1}{\sqrt{E}}\text{ and }E<\frac{1}{x^{2}}\text{ and }E>0\text{ and }x<0\right)\text{ or }\left(x\neq 0\text{ and }E=0\right)\text{ or }\left(x\neq -\sqrt{-\frac{1}{E}}\text{ and }x>-\frac{1}{\sqrt{-E}}\text{ and }E>-\frac{1}{x^{2}}\text{ and }E<0\text{ and }x<0\right)\\L=\frac{Ex^{3}}{\sqrt{1-E^{2}x^{4}}}\text{, }&\left(x<\frac{1}{\sqrt{E}}\text{ and }E<\frac{1}{x^{2}}\text{ and }E>0\text{ and }x>0\right)\text{ or }\left(x\neq 0\text{ and }E=0\right)\text{ or }\left(x\neq \sqrt{-\frac{1}{E}}\text{ and }x<\frac{1}{\sqrt{-E}}\text{ and }E>-\frac{1}{x^{2}}\text{ and }E<0\text{ and }x>0\right)\end{matrix}\right.$$