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Formula
Calculate the integral
$$\displaystyle\int { \dfrac { 1 } { \sin\left( x \right) } } d { x }$$
$- 1 \times \left ( - 1 \right ) \left ( \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) - 1 | \right) } - \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) + 1 | \right) } \right )$
Calculate the integral
$\displaystyle\int { \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ \sin\left( \color{#FF6800}{ x } \right) } } } } d { \color{#FF6800}{ x } }$
 Substitute with $u = \cos\left( x \right)$ and calculate the integral 
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \cos\left( \color{#FF6800}{ x } \right) } }$
$\left [ \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \cos\left( x \right) }$
 It is $\int c f(x) dx = c \int f(x) dx$
$\left [ \color{#FF6800}{ - } \color{#FF6800}{ 1 } \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \cos\left( x \right) }$
 Make the coefficient of the leading term in the denominator 1. 
$\left [ - 1 \times \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \times \frac { 1 } { - 1 } \displaystyle\int { \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } } d { \color{#FF6800}{ u } } \right ] _ { u = \cos\left( x \right) }$
 Calculate the integral using partial integration 
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \left [ \displaystyle\int { \color{#FF6800}{ v } ^ { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } d { \color{#FF6800}{ v } } \right ] _ { \color{#FF6800}{ v } = \color{#FF6800}{ u } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } \color{#FF6800}{ - } \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \left [ \displaystyle\int { \color{#FF6800}{ v } ^ { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } d { \color{#FF6800}{ v } } \right ] _ { \color{#FF6800}{ v } = \color{#FF6800}{ u } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ) \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \left [ \displaystyle\int { \color{#FF6800}{ v } ^ { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } d { \color{#FF6800}{ v } } \right ] _ { v = u - 1 } - \frac { 1 } { 2 } \left [ \displaystyle\int { v ^ { - 1 } } d { v } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
 Calculate the integral using the formula of $\int x^{-1} dx = \ln(|x|)$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \left [ \ln { \left( | \color{#FF6800}{ v } | \right) } \right ] _ { v = u - 1 } - \frac { 1 } { 2 } \left [ \displaystyle\int { v ^ { - 1 } } d { v } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \left [ \ln { \left( | \color{#FF6800}{ v } | \right) } \right ] _ { \color{#FF6800}{ v } = \color{#FF6800}{ u } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } - \frac { 1 } { 2 } \left [ \displaystyle\int { v ^ { - 1 } } d { v } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
 Return the substituted value 
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \ln { \left( | \color{#FF6800}{ u } \color{#FF6800}{ - } \color{#FF6800}{ 1 } | \right) } - \frac { 1 } { 2 } \left [ \displaystyle\int { v ^ { - 1 } } d { v } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \ln { \left( | u - 1 | \right) } - \frac { 1 } { 2 } \left [ \displaystyle\int { \color{#FF6800}{ v } ^ { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } d { \color{#FF6800}{ v } } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
 Calculate the integral using the formula of $\int x^{-1} dx = \ln(|x|)$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \ln { \left( | u - 1 | \right) } - \frac { 1 } { 2 } \left [ \ln { \left( | \color{#FF6800}{ v } | \right) } \right ] _ { v = u + 1 } \right ) \right ] _ { u = \cos\left( x \right) }$
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \ln { \left( | u - 1 | \right) } - \frac { 1 } { 2 } \left [ \ln { \left( | \color{#FF6800}{ v } | \right) } \right ] _ { \color{#FF6800}{ v } = \color{#FF6800}{ u } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ) \right ] _ { u = \cos\left( x \right) }$
 Return the substituted value 
$\left [ - 1 \times \frac { 1 } { - 1 } \left ( \frac { 1 } { 2 } \ln { \left( | u - 1 | \right) } - \frac { 1 } { 2 } \ln { \left( | \color{#FF6800}{ u } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ) \right ] _ { u = \cos\left( x \right) }$
$\left [ \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } \left ( \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } \color{#FF6800}{ - } \color{#FF6800}{ 1 } | \right) } \color{#FF6800}{ - } \color{#FF6800}{ \frac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ u } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right ) \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \cos\left( \color{#FF6800}{ x } \right) } }$
 Return the substituted value 
$\color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ \cos\left( \color{#FF6800}{ x } \right) } \color{#FF6800}{ - } \color{#FF6800}{ 1 } | \right) } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \ln { \left( | \color{#FF6800}{ \cos\left( \color{#FF6800}{ x } \right) } \color{#FF6800}{ + } \color{#FF6800}{ 1 } | \right) } \right )$
$- 1 \times \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } \left ( \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) - 1 | \right) } - \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) + 1 | \right) } \right )$
 Calculate the value 
$- 1 \times \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \left ( \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) - 1 | \right) } - \dfrac { 1 } { 2 } \ln { \left( | \cos\left( x \right) + 1 | \right) } \right )$
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