int_0^1 frac{x e^x}{(2+x)^3} d x is equal to | vedclass

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(D) To evaluate $I = \int_0^1 \frac{x e^x}{(2+x)^3} d x$,we rewrite the integrand as follows:$\frac{x}{(2+x)^3} = \frac{(x+2)-2}{(2+x)^3} = \frac{1}{(

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$\frac{1}{9} \cdot e-\frac{1}{4}$

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(D) To evaluate $I = \int_0^1 \frac{x e^x}{(2+x)^3} d x$,we rewrite the integrand as follows:$\frac{x}{(2+x)^3} = \frac{(x+2)-2}{(2+x)^3} = \frac{1}{(2+x)^2} - \frac{2}{(2+x)^3}$.Let $f(x) = \frac{1}{(2+x)^2}$. Then $f'(x) = -2(2+x)^{-3} = -\frac{2}{(2+x)^3}$.Thus,the integral becomes $\int_0^1 e^x [f(x) + f'(x)] d x$.Using the standard result $\int e^x [f(x) + f'(x)] d x = e^x f(x) + C$,we get:$I = [e^x \cdot \frac{1}{(2+x)^2}]_0^1$.Evaluating at the limits:$I = (e^1 \cdot \frac{1}{(2+1)^2}) - (e^0 \cdot \frac{1}{(2+0)^2}) = \frac{e}{9} - \frac{1}{4}$.

$\int_0^1 \frac{x e^x}{(2+x)^3} d x$ is equal to

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