Question

Integrate:

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

Answer 1

We have,

$I=\int \frac{x}{e^{x^2}}dx$

Let $t=x^2$

$\frac{dt}{dx}=2x$

$\frac{dt}{2}=xdx$

Therefore,

$I=\frac{1}{2}\int \frac{dt}{e^t}$

$I=\frac{1}{2}\int e^{-t}dt$

$I=\frac{1}{2}\left(\frac{e^{-t}}{-1}\right)+C$

$I=-\frac{e^{-t}}{2}+C$

$I=-\frac{1}{2e^t}+C$

On putting the value of $t$, we get

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

Hence, this is the answer.