Mathematics

# Integrate:$\frac { 3 x + 5 } { ( x + 1 ) ( x - 2 ) ^ { 2 } } d x$

##### SOLUTION
Let  $\dfrac { 3x+5 }{ \left( x+1 \right) { \left( x-2 \right) }^{ 2 } } =\dfrac { A }{ x+1 } +\dfrac { B }{ x-2 } +\dfrac { C }{ { \left( x-2 \right) }^{ 2 } }$
On comparing, we get
$A=\dfrac { 2 }{ 9 }$,  $B=\dfrac { -2 }{ 9 }$,  $C=\dfrac { 11 }{ 3 }$
So,  $\int { \dfrac { \left( 3x+5 \right) }{ \left( x+1 \right) { \left( x-2 \right) }^{ 2 } } dx } =\int { \dfrac { 2 }{ 9\left( x+1 \right) } dx } +\int { \dfrac { -2 }{ 9\left( x-2 \right) } dx } +\int { \dfrac { 11 }{ 3{ \left( x-2 \right) }^{ 2 } } dx }$
So,  $\int { \dfrac { 3x+5 }{ \left( x+1 \right) { \left( x-2 \right) }^{ 2 } } } =\dfrac { 2 }{ 9 } log\left| x+1 \right| -\dfrac { 2 }{ 9 } ln\left( x-2 \right) -\dfrac { 11 }{ 3 } \dfrac { 1 }{ \left( x-2 \right) } +C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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