Solve A = ∫ _{-sqrt{0.5}}^u2x^2+tx+t/x wrt x

$$A = \int_{ - \sqrt{ 0.5 } }^{ u } \frac{ 2 { x }^{ 2 } +tx+t }{ x } d x$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{\sqrt{2}x\left(2ux+\sqrt{2}x-A\right)}{\sqrt{2}ux+\sqrt{2}u+x+1}\text{, }&u\neq -\frac{\sqrt{2}}{2}\text{ and }x\neq -1\text{ and }x\neq 0\text{ and }u\neq 0\\t\in \mathrm{R}\text{, }&\left(u=\frac{-A-\sqrt{2}}{2}\text{ and }A\neq 0\text{ and }A\neq -\sqrt{2}\text{ and }x=-1\right)\text{ or }\left(u=-\frac{\sqrt{2}}{2}\text{ and }x\neq 0\text{ and }A=0\right)\end{matrix}\right.$$