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Q.
The value of $\int x^2 e^x d x$ is
Integrals
Solution:
Let $I=\int x^2 e^x d x$
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 d x-\int\left[\frac{d}{d x}\left(x^2\right) \int e^x d x\right] d x $
$ =x^2 e^x-\int\left[2 x e^x\right] d x$
Again, integrating by parts, we get
$ I=x^2 e^x-\left\{2 x \int e^x d x-2 \int\left[\frac{d}{d x}(x) \int e^x d x\right] d x\right\}$
$=x^2 e^x-2 x e^x+2 \int e^x d x$
$=x^2 e^x-2 x e^x+2 e^x+C$
$I=e^x\left(x^2-2 x+2\right)+C$