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Formula
Calculate the integral
$\int{ x \sin\left( x \right) }d{ x }$
$1 \left ( - x \cos\left( x \right) + \sin\left( x \right) \right )$
Calculate the integral
$\displaystyle\int { \color{#FF6800}{ x } \color{#FF6800}{ \sin\left( x \right) } } d { \color{#FF6800}{ x } }$
 Calculate the integral using the formula of $\int x \sin^{n}(x) dx =\dfrac{1}{n}(-x\cos(x)\sin^{n-1}(x)+\int{\cos(x)\sin^{n-1}(x)}d{x}+(n-1)\int{x\sin^{n-2}(x)}d{x})$
$\color{#FF6800}{ \dfrac { 1 } { 1 } } \left ( \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ \cos\left( x \right) } \color{#FF6800}{ \sin ^ { 1 - 1 } \left ( x \right) } \color{#FF6800}{ + } \displaystyle\int { \color{#FF6800}{ \cos\left( x \right) } \color{#FF6800}{ \sin ^ { 1 - 1 } \left ( x \right) } } d { \color{#FF6800}{ x } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \displaystyle\int { \color{#FF6800}{ x } \color{#FF6800}{ \sin ^ { 1 - 2 } \left ( x \right) } } d { \color{#FF6800}{ x } } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) \sin ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } \left ( x \right) + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Add $1$ and $- 1$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) \sin ^ { \color{#FF6800}{ 0 } } \left ( x \right) + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) \color{#FF6800}{ \sin ^ { 0 } \left ( x \right) } + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Calculate power 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) \times \color{#FF6800}{ 1 } + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ \cos\left( x \right) } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Multiplying any number by 1 does not change the value 
$\dfrac { 1 } { 1 } \left ( \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ \cos\left( x \right) } + \displaystyle\int { \cos\left( x \right) \sin ^ { 1 - 1 } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \displaystyle\int { \cos\left( x \right) \sin ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Add $1$ and $- 1$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \displaystyle\int { \cos\left( x \right) \sin ^ { \color{#FF6800}{ 0 } } \left ( x \right) } d { x } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \displaystyle\int { \color{#FF6800}{ \cos\left( x \right) } \color{#FF6800}{ \sin ^ { 0 } \left ( x \right) } } d { \color{#FF6800}{ x } } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Substitute with $u = \sin\left( x \right)$ and calculate the integral 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \left [ \displaystyle\int { \color{#FF6800}{ 1 } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \sin\left( x \right) } } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \left [ \displaystyle\int { \color{#FF6800}{ 1 } } d { \color{#FF6800}{ u } } \right ] _ { u = \sin\left( x \right) } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 The indefinite integral of $1$ is $x$ . 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \left [ \color{#FF6800}{ u } \right ] _ { u = \sin\left( x \right) } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \left [ \color{#FF6800}{ u } \right ] _ { \color{#FF6800}{ u } = \color{#FF6800}{ \sin\left( x \right) } } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Return the substituted value 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \color{#FF6800}{ \sin\left( x \right) } + \left ( 1 - 1 \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) + \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 Add $1$ and $- 1$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) + \color{#FF6800}{ 0 } \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) + \color{#FF6800}{ 0 } \displaystyle\int { x \sin ^ { 1 - 2 } \left ( x \right) } d { x } \right )$
 If you multiply a number by 0, it becomes 0 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) + \color{#FF6800}{ 0 } \right )$
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) \color{#FF6800}{ + } \color{#FF6800}{ 0 } \right )$
 0 does not change when you add or subtract 
$\dfrac { 1 } { 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) \right )$
$\color{#FF6800}{ \dfrac { 1 } { 1 } } \left ( - x \cos\left( x \right) + \sin\left( x \right) \right )$
 Calculate the value 
$\color{#FF6800}{ 1 } \left ( - x \cos\left( x \right) + \sin\left( x \right) \right )$
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