Question

integral de x^3 e^-x^2 dx

Answer 1

$\displaystyle I=\int x^3e^{-x^2}dx\\ \\ \texttt{Cambio de variable: }u=x^2\to du = 2x~dx\to x~dx =\dfrac{1}{2}~du\\ \\ \\ I=\int x^2e^{-x^2}\cdot x~dx\\ \\ I=\int u{e^{-u}}\cdot \dfrac{1}{2}~du\\ \\ I=\dfrac{1}{2}\int u{e^{-u}}~du\\ \\ \texttt{Aprovechando que }d(e^{-u})=-e^{-u}~du\to e^{-u}~du=-d(e^{-u})\\ \\ I=-\dfrac{1}{2}\int u~d(e^{-u})\\ \\ \\ I=-\dfrac{1}{2}\left[ue^{-u}-\int e^{-u}~du\right]\\ \\ \\ I=-\dfrac{1}{2}\left[ue^{-u}+e^{-u}\right]+C\\ \\ \\ I=-\dfrac{e^{-u}(u+1)}{2}+C$


$\texttt{Devolviendo variable:}\\ \\ \hspace*{3cm}\boxed{I=-\dfrac{e^{-x^2}(x^2+1)}{2}+C}$