Question

$\underset{0}{\overset{1}{\int }}x{e}^{x^2}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_0^{1}x{e}^{x^2}\mathit{d}x. \\ \mathrm{Let}{x}^2=t.\mathrm{Then},2xdx=dt \\ \mathrm{When}x=0,t=0\mathrm{and}x=1,t=1 \\ \therefore I=\frac{1}{2}{\int }_0^1{e}^{t}dt \\ \Rightarrow I=\frac{1}{2}{\left(e^t\right)}_0^1 \\ \Rightarrow I=\frac{1}{2}\left(e-1\right) \end{gathered}$$