Integrate the functions in Exercises: x e^ x^2 - Vidyadip

Question

Integrate the functions in Exercises:
$\frac{\text{x}}{\text{e}^{\text{x}^2}}$

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Answer

$\text{Let I}=\int\frac{\text{x}}{\text{e}^{\text{x}^{2}}} \text{ dx}=\frac{1}{2}\int\frac{\text{2x}}{\text{e}^{\text{x}^{2}}}\text{ dx}\ \ \ \ ....\text{(i)} $
$\text{Putting}\text{ x}^2=\text{t}\ \ \ \ \Rightarrow \ \ \ \ 2\text{x}=\frac{\text{dt}}{\text{dx}}\ \ \ \ \Rightarrow \ \ \ \ \ \text{2x}{\text{ dx}}=\text{ dt} $
$\therefore \ \ \ \ $From eq. (i), $\text{I}=\frac{1}{2}\int\frac{\text{dt}}{\text{e}^{\text{t}}}=\frac{1}{2}\int\text{e}^{\text{-t}}\text{ dt}$
$=\frac{1}{2}\cdot\frac{\text{e}^{-\text{t}}}{-1\rightarrow \text{Coeff. of t}}+\text{c}$
$=\frac{-1}{2(\text{e}^{\text{t}})}+\text{c}=\frac{-1}{2(\text{e}^{\text{x}^{2}})}+\text{c} $

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