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Question
\[\frac{dy}{dx} = x^2 e^x\]
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Solution
We have,
\[\frac{dy}{dx} = x^2 e^x \]
\[ \Rightarrow dy = x^2 e^x dx\]
Integrating both sides, we get
\[ \Rightarrow \int dy = x^2 \int e^x dx - \int\left( \frac{d}{dx}\left( x^2 \right)\int e^x dx \right)dx\]
\[ \Rightarrow y = x^2 e^x - 2\int x e^x dx\]
\[ \Rightarrow y = x^2 e^x - 2x\int e^x dx + 2\int\left( \frac{d}{dx}\left( x \right)\int e^x dx \right)dx\]
\[ \Rightarrow y = x^2 e^x - 2x e^x + 2 e^x + C\]
\[ \Rightarrow y = \left( x^2 - 2x + 2 \right) e^x + C\]
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