Solve ∫ (from 0 to 100) of x^{2}+2xe^{4 wrt x+3x^{3}+2x^2+a^2+3^2=10^2n+2

$$\int _ { 0 } ^ { 100 } x ^ { 2 } + 2 x e ^ { 4 } d x + 3 x ^ { 3 } + 2 x ^ { 2 } + a ^ { 2 } + 3 ^ { 2 } = 10 ^ { 2 n + 2 }$$

Solve for a

$$a=\frac{\sqrt{-27x^{3}-18x^{2}+900\times 100^{n}-90000e^{4}-3000081}}{3}$$

$$a=-\frac{\sqrt{-27x^{3}-18x^{2}+900\times 100^{n}-90000e^{4}-3000081}}{3}\text{, }-3x^{3}-2x^{2}+100^{n+1}-10000e^{4}-\frac{1000027}{3}\geq 0$$

Solve for n

$$n=\frac{\log(\frac{9x^{3}+6x^{2}+3a^{2}+30000e^{4}+1000027}{3})-2}{2}$$

$$-\left(3x^{3}+2x^{2}+10000e^{4}+\frac{1000027}{3}\right)<0\text{ or }\left(-\left(3x^{3}+2x^{2}+10000e^{4}+\frac{1000027}{3}\right)\geq 0\text{ and }a>\frac{\sqrt{-27x^{3}-18x^{2}-90000e^{4}-3000081}}{3}\text{ and }-27x^{3}-18x^{2}-90000e^{4}-3000081\geq 0\text{ and }a>\sqrt{-\left(3x^{3}+2x^{2}+10000e^{4}+\frac{1000027}{3}\right)}\right)\text{ or }\left(-\left(3x^{3}+2x^{2}+10000e^{4}+\frac{1000027}{3}\right)\geq 0\text{ and }a<-\frac{\sqrt{-27x^{3}-18x^{2}-90000e^{4}-3000081}}{3}\text{ and }-27x^{3}-18x^{2}-90000e^{4}-3000081\geq 0\text{ and }a<-\sqrt{-\left(3x^{3}+2x^{2}+10000e^{4}+\frac{1000027}{3}\right)}\right)$$