Question

$\frac{dy}{dx}=x^2{e}^x$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}=x^2{e}^x \\ \Rightarrow dy=x^2{e}^{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int dy=\int \underset{\mathrm{I}}{x^2}\underset{\mathrm{II}}{e^x}dx \\ \Rightarrow \int dy=x^2\int e^{x}dx-\int \left(\frac{d}{dx}\left(x^2\right)\int e^{x}dx\right)dx \\ \Rightarrow y=x^2{e}^x-2\int x{e}^{x}dx \\ \Rightarrow y=x^2{e}^x-2\int \underset{\mathrm{I}}{x}\underset{\mathrm{II}}{e^x}dx \\ \Rightarrow y=x^2{e}^x-2x\int e^{x}dx+2\int \left(\frac{d}{dx}\left(x\right)\int e^{x}dx\right)dx \\ \Rightarrow y=x^2{e}^x-2x{e}^x+2e^x+C \\ \Rightarrow y=\left(x^2-2x+2\right)e^x+C \end{gathered}$$