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