Please revise the solution

Unfortunately, I don't think students learn the fancy formula in the current solution. Although they know this, it would be more educational to explain based on the fundamental theorem of calculus and the chain rule. I hope the solution is revised as follows.

$\int _{1-x}^{1+2x}e^{t^{2}}dt=\int _{1-x}^{0}e^{t^{2}}dt+\int _{0}^{1+2x}e^{t^{2}}dt=-\int _{0}^{1-x}e^{t^{2}}dt+\int _{0}^{1+2x}e^{t^{2}}dt$

Define $g(y)=\int _{0}^{y}e^{t^{2}}dt$. Then, the integral would be $-g(1-x)+g(1+2x)$.

Then, use the fundamental theorem of calculus and the chain rule....