The value of int_0^1 {x^2 e^x dx} is equal to | vedclass

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(A) Let $I = \int_0^1 x^2 e^x dx$. Using integration by parts,$\int u v dx = u \int v dx - \int (u' \int v dx) dx$. Let $u = x^2$ and $v = e^x$. Then

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$e - 2$

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(A) Let $I = \int_0^1 x^2 e^x dx$. Using integration by parts,$\int u v dx = u \int v dx - \int (u' \int v dx) dx$. Let $u = x^2$ and $v = e^x$. Then $u' = 2x$ and $\int v dx = e^x$. $I = [x^2 e^x]_0^1 - \int_0^1 2x e^x dx$. $I = (1^2 e^1 - 0^2 e^0) - 2 \int_0^1 x e^x dx$. $I = e - 2 \int_0^1 x e^x dx$. Now,evaluate $\int_0^1 x e^x dx$ using integration by parts again: $\int_0^1 x e^x dx = [x e^x]_0^1 - \int_0^1 e^x dx = (e - 0) - [e^x]_0^1 = e - (e - 1) = 1$. Substituting this back into the expression for $I$: $I = e - 2(1) = e - 2$.

The value of $\int_0^1 {x^2 e^x dx}$ is equal to

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