Question

$\int 2x^3{e}^{x^2}dx$

Answer 1

$$\begin{gathered} \int 2x^3·e^{x^2}dx \\ =\int x^2·\left(e^{x^2}\right)·2xdx \\ L\mathrm{et}{x}^2=t \\ \Rightarrow 2xdx=dt \\ =\int \underset{\mathrm{I}}{t}·{\underset{\mathrm{II}}{e}}^{t}dt \\ =t·e^t-\int 1·e^{t}dt \\ =t{e}^t-e^t+C \\ =x^2{e}^{x^2}-e^{x^2}+C \\ \mathit{=}{e}^{{\mathit{x}}^{\mathit{2}}}\left({\mathit{x}}^{\mathit{2}}\mathit{-}\mathit{1}\right)\mathit{+}C \end{gathered}$$