Mathematics

# $\int \dfrac { 8 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) } d x$ is equal to

$\operatorname { log } | x + 2 | - \dfrac { 1 } { 2 } \log \left( x ^ { 2 } + 4 \right) + \tan ^ { - 1 } \dfrac { x } { 2 } + C$

##### SOLUTION
We have,
$I=\int { \dfrac { 8 }{ { \left( { x+2 } \right) \left( { { x^{ 2 } }+4 } \right) } } fd }$
$I=\dfrac { 8 }{ { \left( { x+2 } \right) \left( { { x^{ 2 } }+4 } \right) } }$
$I=\dfrac { A }{ { x+2 } } +\dfrac { { Bx+C } }{ { { x^{ 2 } }+4 } }$

On solving above expression, we get
$A=1, B=2, C=-1$

Therefore,
$I=\int { \dfrac { 8 }{ { \left( { x+2 } \right) \left( { { x^{ 2 } }+4 } \right) } } dx } \\ I=\int { \dfrac { { dx } }{ { x+2 } } +\int { \dfrac { { 2-x } }{ { { x^{ 2 } }+4 } } dx } } \\ I=\log |x+2|+\dfrac { 2 }{ 2 } { \tan ^{ -1 } }\left( { \dfrac { x }{ 2 } } \right) -\dfrac { 1 }{ 2 } \log \left( { { x^{ 2 } }+4 } \right) +C \\I= \log |x+2|+{ \tan ^{ -1 } }\left( { \dfrac { x }{ 2 } } \right) -\dfrac { 1 }{ 2 } \log \left( { { x^{ 2 } }+4 } \right) +C$

Hence, this is the correct answer.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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#### Realted Questions

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