\(\int \frac{x^{2 x}}{(1+2 x)} d x=\)
(where C is a constant of integration.)

\(\int \frac{x \cdot e^{2 x}}{(1+2 x)^2} d x\)
Let \(2 \mathrm{x}=\mathrm{t} \Rightarrow 2 \mathrm{dx}=\mathrm{dt}\)
\(=\frac{1}{4} \int \frac{2 \mathrm{x} \cdot \mathrm{e}^{2 \mathrm{x}}}{(1+2 \mathrm{x})^2} \cdot 2 \mathrm{dx}\)
\(=\frac{1}{4} \int \frac{\mathrm{te} \mathrm{dt}}{(1+\mathrm{t})^2}\)
\(=\frac{1}{4} \int \mathrm{e}^{\mathrm{t}}\left\{\frac{1}{1+\mathrm{t}}-\frac{1}{(1+\mathrm{t})^2}\right\} \mathrm{dt}\)
\(=\frac{1}{4} \mathrm{e}^{\mathrm{t}} \cdot \frac{1}{1+\mathrm{t}}+\mathrm{C}[\because \int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=\) \(\mathrm{e}^{\mathrm{xf}}(\mathrm{x})+\mathrm{C}]\)