Mathematics

Find :
$$\int {\frac{e^x(x-1)}{(x+1)^3}dx}$$


SOLUTION
Find 
$$\displaystyle\int { { e }^{ x } } \dfrac { \left( x-1 \right)  }{ { \left( x-1 \right)  }^{ 3 } } dx$$
$$\displaystyle\int { { e }^{ x } } \dfrac { x+1-2 }{ { \left( x-1 \right)  }^{ 3 } } dx$$
$$\displaystyle\int { { e }^{ x } } \left\{ \dfrac { x+1 }{ { \left( x+1 \right)  }^{ 3 } } -\dfrac { 2 }{ { \left( x+1 \right)  }^{ 3 } }  \right\} dx$$
$$ =\displaystyle\int { \left\{ \dfrac { { e }^{ x } }{ { \left( x+1 \right)  }^{ 2 } } -\dfrac { 2{ e }^{ x } }{ { \left( x+1 \right)  }^{ 3 } }  \right\} dx }$$
$$ \Rightarrow \dfrac { { e }^{ x }{ \left( x+1 \right)  }^{ 2 }-{ e }^{ x }2\left( x+1 \right) dx }{ { \left( x+1 \right)  }^{ 4 } } =dz$$
Or $$ \left\{ \dfrac { { e }^{ x } }{ { \left( x+1 \right)  }^{ 2 } } -\dfrac { 2{ e }^{ x } }{ { \left( x+1 \right)  }^{ 3 } }  \right\} dx=dz$$
$$ \displaystyle\int { dz=z+c } $$
$$=\dfrac { { e }^{ x } }{ { \left( x+1 \right)  }^{ 2 } } +c$$
Ans $$=\dfrac { { e }^{ x } }{ { \left( x+1 \right)  }^{ 2 } }$$

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Subjective Medium Published on 17th 09, 2020
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