Answer
$\frac{e-2}{8}$
Work Step by Step
Given $$\int_0^{1/2} \frac{x e^{2 x}}{(1+2 x)^2} d x$$
Integrate by parts
\begin{aligned}
u&= xe^{2x}\ \ \ \ \ \ \ \ \ \ \ &dv& = \frac{dx }{(1+2 x)^2}\\
du&= (2xe^{2x}+e^{2x})dx\ \ \ \ \ \ \ \ \ \ \ & v&= \frac{-1}{2}\frac{1}{1+2x}
\end{aligned}
Then
\begin{aligned}
\int _0^{1/2} \frac{x e^{2 x}}{(1+2 x)^2} d x&= -\frac{1}{2} \cdot \frac{x e^{2 x}}{1+2 x}\bigg|_0^{1/2} -\int_0^{1/2} \left[-\frac{1}{2} \cdot \frac{e^{2 x}(2 x+1)}{1+2 x}\right] d x\\
&=-\frac{x e^{2 x}}{4 x+2}+\frac{1}{2} \cdot \frac{1}{2} e^{2 x}\bigg|_0^{1/2} \\
&= \frac{1}{8} e-\frac{1}{4}\\
&=\frac{e-2}{8}
\end{aligned}
