Question

integrate the function [x/e^x²].dx

Answer 1

$\bf{Let,I=\int{\frac{x}{e^{x^2}}}\,dx}\\\bf{\int{xe^{-x^2}}\,dx}$

Let -x² = t
differentiate both sides,
-2x.dx = dt
xdx = -1/2 dt , put it in I
so, I = $\bf{\int{e^t\frac{-1}{2}}\,dt}$

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

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

put t = -x²
so, I = $\bf{\frac{-1}{2}e^{-x^2}+C}$