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Formula
Calculate the integral
$$\displaystyle\int { \tan\left( x \right) } d { x }$$
$\dfrac { 1 } { 2 } \ln { \left( | \tan ^ { 2 } \left ( x \right) + 1 | \right) }$
Calculate the integral
$\displaystyle\int { \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } } d { \color{#FF6800}{ x } }$
 Substitute with $u = \tan\left( x \right)$ and calculate the integral 
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ u } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } }$
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ u } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \tan\left( x \right) }$
 Calculate the integral when the numerator divides by the derivative of the denominator. 
$\left [ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ] _ { u = \tan\left( x \right) }$
$\left [ \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \tan\left( \color{#FF6800}{ x } \right) } }$
 Return the substituted value 
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ \tan ^ { \color{#FF6800}{ 2 } } \left ( \color{#FF6800}{ x } \right) } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) }$
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