int_{0}^{2} x e^{x} dx = | vedclass

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(A) To evaluate the integral $\int_{0}^{2} x e^{x} dx$,we use the method of integration by parts: $\int u dv = uv - \int v du$.Let $u = x$ and $dv = e

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$e^{2} + 1$

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(A) To evaluate the integral $\int_{0}^{2} x e^{x} dx$,we use the method of integration by parts: $\int u dv = uv - \int v du$.Let $u = x$ and $dv = e^{x} dx$. Then $du = dx$ and $v = e^{x}$.Applying the formula: $\int x e^{x} dx = x e^{x} - \int e^{x} dx = x e^{x} - e^{x}$.Now,applying the limits from $0$ to $2$:$\left[ x e^{x} - e^{x} \right]_{0}^{2} = (2 e^{2} - e^{2}) - (0 \cdot e^{0} - e^{0})$.$= (e^{2}) - (0 - 1) = e^{2} + 1$.

$\int_{0}^{2} x e^{x} dx =$

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