Question

Integrate the following functions.
$\int \frac{x}{e^{x^2}}dx.$

Answer 1

$\int \frac{x}{e^{x^2}}dx$
Let $x^2=t\Rightarrow 2x=\frac{dt}{dx}\to dx=\frac{dt}{2x}$
$\therefore \int \frac{x}{e^{x^2}}dx=\int \frac{x}{e^t}\frac{dt}{2x}=\frac{1}{2}\int e^{-t}dt=-\frac{1}{2}{e}^{-t}+C=-\frac{1}{2}{e}^{-\left(x^2\right)}+C$