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(B) To evaluate the integral $\int x e^{x} d x$,we use the method of integration by parts,which states that $\int u v d x = u \int v d x - \int (\frac

Vedclass · Accepted Answer

$x e^{x} - e^{x} + C$

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(B) To evaluate the integral $\int x e^{x} d x$,we use the method of integration by parts,which states that $\int u v d x = u \int v d x - \int (\frac{du}{dx} \int v d x) d x$.Let $u = x$ and $v = e^{x}$.Then $\frac{du}{dx} = 1$ and $\int v d x = e^{x}$.Applying the formula:$\int x e^{x} d x = x e^{x} - \int (1 \cdot e^{x}) d x$$= x e^{x} - \int e^{x} d x$$= x e^{x} - e^{x} + C$$= e^{x}(x - 1) + C$

Find $\int x e^{x} d x$

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