x-2-e-x-2-2-Mastermind-

Question Number 169049 by Mastermind last updated on 23/Apr/22

\int x^{2} e^{\frac{x^{2}}{2}}

M a s t e r m i n d

Answered by Mathspace last updated on 23/Apr/22

\int x^{2} e^{\frac{x^{2}}{2}} d x = \int (x) ({x e}^{\frac{x^{2}}{2}}) d x

= {x e}^{\frac{x^{2}}{2}} - \int e^{\frac{x^{2}}{2}} d x

= {x e}^{\frac{x^{2}}{2}} - \int \sum_{n = 0}^{\infty} \frac{{(\frac{x^{2}}{2})}^{n}}{n!}

= {x e}^{\frac{x^{2}}{2}} - \sum_{n = 0}^{\infty} \frac{1}{2^{n} n!} \int x^{2 n} d x

= {x e}^{\frac{x^{2}}{2}} - \sum_{n = 0}^{\infty} \frac{x^{2 n + 1}}{n! 2^{n} (2 n + 1)} + C