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Calculate the integral
Answer
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$\int{ 5 ^{ x } }d{ x }$
$\log _{ 5 } { \left( e \right) } e ^ { \ln { \left( 5 \right) } x }$
Calculate the integral
$\displaystyle\int { \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$ $ Calculate the integral using the formula of $ \int a^{x} dx = \int e^{\ln(a) x} dx$
$\displaystyle\int { \color{#FF6800}{ e } ^ { \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$\displaystyle\int { \color{#FF6800}{ e } ^ { \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } } } d { \color{#FF6800}{ x } }$
$ $ Substitute with $ u = \ln { \left( 5 \right) } x $ and calculate the integral $ $
$\left [ \color{#FF6800}{ \frac { 1 } { \ln { \left( 5 \right) } } } \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } }$
$\left [ \frac { 1 } { \ln { \left( 5 \right) } } \displaystyle\int { \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } } d { \color{#FF6800}{ u } } \right ] _ { u = \ln { \left( 5 \right) } x }$
$ $ Calculate the integral using the formula of $ \int e^{x} dx = e^{x}$
$\left [ \frac { 1 } { \ln { \left( 5 \right) } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } \right ] _ { u = \ln { \left( 5 \right) } x }$
$\left [ \color{#FF6800}{ \frac { 1 } { \ln { \left( 5 \right) } } } \color{#FF6800}{ e } ^ { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } }$
$ $ Return the substituted value $ $
$\color{#FF6800}{ \dfrac { 1 } { \ln { \left( 5 \right) } } } \color{#FF6800}{ e } ^ { \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } }$
$\color{#FF6800}{ \dfrac { 1 } { \ln { \left( 5 \right) } } } \color{#FF6800}{ e } ^ { \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } }$
$ $ Simplify the expression $ $
$\log _{ \color{#FF6800}{ 5 } } { \left( \color{#FF6800}{ e } \right) } \color{#FF6800}{ e } ^ { \ln { \left( \color{#FF6800}{ 5 } \right) } \color{#FF6800}{ x } }$
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