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Calculate the integral
Answer
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$\int{ xe ^{ -x } }d{ x }$
$- x \times \dfrac { 1 } { e ^ { x } } - \dfrac { 1 } { e ^ { x } }$
Calculate the integral
$\displaystyle\int { \color{#FF6800}{ x } \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$ $ Calculate the integral using partial integration $ $
$\color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } \color{#FF6800}{ + } \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$- x e ^ { - x } + \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$ $ Use the integration by substitution to calculate the integral $ $
$- x e ^ { - x } + \left [ \color{#FF6800}{ - } \displaystyle\int { \color{#FF6800}{ \frac { 1 } { u } } \color{#FF6800}{ u } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } }$
$- x e ^ { - x } + \left [ - \displaystyle\int { \color{#FF6800}{ \frac { 1 } { u } } \color{#FF6800}{ u } } d { \color{#FF6800}{ u } } \right ] _ { u = e ^ { - x } }$
$ $ Calculate the differentiation of the logarithmic function $ $
$- x e ^ { - x } + \left [ - \color{#FF6800}{ u } \right ] _ { u = e ^ { - x } }$
$- x e ^ { - x } + \left [ \color{#FF6800}{ - } \color{#FF6800}{ u } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } }$
$ $ Return the substituted value $ $
$- x e ^ { - x } \color{#FF6800}{ - } \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } }$
$\color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } } \color{#FF6800}{ - } \color{#FF6800}{ e } ^ { \color{#FF6800}{ - } \color{#FF6800}{ x } }$
$ $ Simplify the expression $ $
$\color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 1 } { e ^ { x } } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { e ^ { x } } }$
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