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Formula
Calculate the integral
Answer
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$\int{ xe ^{ x ^{ 2 } } }d{ x }$
$\dfrac { 1 } { 2 } e ^ { x ^ { 2 } } + C$
Calculate the indefinite integral.
$\displaystyle\int { \color{#FF6800}{ x } \color{#FF6800}{ e } ^ { \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } } } d { \color{#FF6800}{ x } }$
$ $ Substitute with $ u = x ^ { 2 } $ and calculate the integral $ $
$\left [ \color{#FF6800}{ \frac { 1 } { 2 } } \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } }$
$\left [ \frac { 1 } { 2 } \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } } d { \color{#FF6800}{ u } } \right ] _ { u = x ^ { 2 } }$
$ $ Calculate the integral using the formula of $ \int e^{x} dx = e^{x}$
$\left [ \frac { 1 } { 2 } \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } \right ] _ { u = x ^ { 2 } }$
$\left [ \color{#FF6800}{ \frac { 1 } { 2 } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } }$
$ $ Return the substituted value $ $
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } }$
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } }$
$ $ Add the integral constant $ C $ . $ $
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } } \color{#FF6800}{ + } \color{#FF6800}{ C }$
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