Consider the definite integral $I={\displaystyle \underset{2}{\overset{5}{}}}{e}^{\sqrt[\phantom{2}]{x^2+1}}dx.$ Using an appropriate change of variables $I={\displaystyle \underset{-1}{\overset{1}{}}}g(t)dt,$ the function $g(t)$ is

$e^{\sqrt[\phantom{2}]{\frac{9}{4}{t}^2+\frac{27}{2}t+\frac{77}{4}}}$

Option $4$

$\frac{3}{2}{e}^{\sqrt[\phantom{2}]{\frac{9}{4}{t}^2+\frac{21}{2}t+\frac{53}{4}}},\frac{3}{2}{e}^{\sqrt[\phantom{2}]{\frac{9}{4}{t}^2+\frac{27}{2}t+\frac{77}{4}}}$

Option $1$

Option $3$