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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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$\displaystyle \int {x+2\sqrt {x-3}}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
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