∫xe^ xdx Integrating by parts,Let u=x⇒ du=dx dv=e^ xdx⇒ v= e^ x = xe^ x ∫ e^ xdx = xe^ x+∫e^ xdx = xe^ x e^ x+c ...

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Area between x=g(y) and y Axis

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Question

Evaluate
$\int x e^{- x} d x$

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Solution

$\int x e^{- x} d x$
Integrating by parts,
Let $u = x \Rightarrow d u = d x$
$d v = e^{- x} d x \Rightarrow v = - e^{- x}$
$= - x e^{- x} - \int - e^{- x} d x$
$= - x e^{- x} + \int e^{- x} d x$
$= - x e^{- x} - e^{- x} + c$

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