Question

The value of $\int x^2{e}^{x}dx$ is

• A. $e^x\left(x^2+2x-2\right)+C$
• B. $e^x\left(x^2-2x-2\right)+C$
• C. $e^x\left(x^2-2x+2\right)+C$
• D. $-e^x\left(x^2+2x+2\right)+C$

Answer 1

Answer: (C)

Let $I=\int x^2{e}^{x}dx$
On taking $x^2$ as first function and $e^x$ as second function and integrating by parts, we get
$I=x^2\int e^{x}dx-\int \left[\frac{d}{dx}\left(x^2\right)\int e^{x}dx\right]dx$
$=x^2{e}^x-\int \left[2x{e}^x\right]dx$
Again, integrating by parts, we get
$I=x^2{e}^x-\left\{2x\int e^{x}dx-2\int \left[\frac{d}{dx}(x)\int e^{x}dx\right]dx\right\}$
$=x^2{e}^x-2x{e}^x+2\int e^{x}dx$
$=x^2{e}^x-2x{e}^x+2e^x+C$
$I=e^x\left(x^2-2x+2\right)+C$