Question

\[\frac{dy}{dx} = x^2 e^x\]

Answer 1

We have,

\[\frac{dy}{dx} = x^2 e^x \]

\[ \Rightarrow dy = x^2 e^x dx\]

Integrating both sides, we get

\[ \Rightarrow \int dy = x^2 \int e^x dx - \int\left( \frac{d}{dx}\left( x^2 \right)\int e^x dx \right)dx\]

\[ \Rightarrow y = x^2 e^x - 2\int x e^x dx\]

\[ \Rightarrow y = x^2 e^x - 2x\int e^x dx + 2\int\left( \frac{d}{dx}\left( x \right)\int e^x dx \right)dx\]

\[ \Rightarrow y = x^2 e^x - 2x e^x + 2 e^x + C\]

\[ \Rightarrow y = \left( x^2 - 2x + 2 \right) e^x + C\]