The Notes: Integration of xe^x^2

$\displaystyle\int{x}{\left({{e}}^{{{{x}}^{{2}}}}\right)}{\left.{d}{x}\right.}$

Let's apply integral substitution,

Substitute $\displaystyle{u}={{x}}^{{2}}$

$\displaystyle\frac{{{d}{u}}}{{\left.{d}{x}}}={2}{x}$

$\displaystyle\Rightarrow{d}{u}={2}{x}{\left.{d}{x}\right.}$

$\displaystyle\Rightarrow{\left.{d}{x}\right.}={\left(\frac{{1}}{{{2}{x}}}\right)}{d}{u}$

$\displaystyle\int{x}{{e}}^{{{{x}}^{{2}}}}{\left.{d}{x}\right.}=\int{x}{{e}}^{{u}}{\left(\frac{{1}}{{{2}{x}}}\right)}{d}{u}$

$\displaystyle=\int\frac{{1}}{{2}}{{e}}^{{u}}{d}{u}$

$\displaystyle=\frac{{1}}{{2}}\int{{e}}^{{u}}{d}{u}$

$\displaystyle=\frac{{1}}{{2}}{{e}}^{{u}}$

Substitute back $\displaystyle{u}={{x}}^{{2}}$

$\displaystyle=\frac{{1}}{{2}}{{e}}^{{{{x}}^{{2}}}}$

Add a constant C to the solution,

$\displaystyle=\frac{{1}}{{2}}{{e}}^{{{{x}}^{{2}}}}+{C}$

The Notes