Mathematics

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


ANSWER

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


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
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