Question

\[\int x e^{2x} \text{ dx }\]

Answer 1

\[\int x e^{2x} \text{ dx }\]
` "Taking x as the first function and      e"^2x  "         as the second function ". `
\[ = x\int e^{2x} dx - \int\left\{ \frac{d}{dx}\left( x \right)\int e^{2x} dx \right\}dx\]
\[ = \frac{x e^{2x}}{2} - \int\left( \frac{e^{2x}}{2} \right)dx\]
\[ = \frac{x}{2} e^{2x} - \frac{e^{2x}}{4} + C\]
\[ = e^{2x} \left( \frac{x}{2} - \frac{1}{4} \right) + C\]